LLemma 34B on 5.NF.B.4
Common Core Standard 5.NF.B.4
This standard covers the following:
Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.
Problem: Determine the product when you multiply the fraction 8/9 by the fraction 19/3. Please provide your answer as a simplified fraction.
LLemma 34B:
Reasoning: 8/9 * 19/3 = <<8/9*19/3=152/27>>152/27 ### Answer: 152/27
Expected answer: 152/27
Correct? Yes
Problem: After determining the product when multiplying the fraction 8/9 by the fraction 19/3, take your result and then multiply it by 2. What does this new calculation result in when written as a simplified fraction?
LLemma 34B:
Reasoning: 152/27 * 2 = <<152/27*2=304/27>>304/27 ### Answer: 304/27
Expected answer: 304/27
Correct? Yes
Problem: Imagine you made an error in the original multiplication calculation where you incorrectly used 8/9 instead of the correct fraction 8/15. Can you correct this mistake and find the new product if you multiply 8/15 by 19/3? Please provide your answer as a simplified fraction.
LLemma 34B:
Reasoning: 8/15 * 19/3 = <<8/15*19/3=28/5>>28/5 ### Answer: 28/5
Expected answer: 152/45
Correct? No
Problem: Julian is cutting sections of rope for a project. Each section is 13/2 feet long. He is only able to use 5/7 of each section. How many feet are in the usable part of each section of rope? Calculate this by multiplying 13/2 by 5/7.
LLemma 34B:
Reasoning: Julian is able to use 5/7 of each section of rope. 5/7 of 13/2 feet is 5/7*13/2 = <<5/7*13/2=13/7>>13/7 feet. ### Answer: 13/7
Expected answer: 65/14
Correct? No
Problem: What is the result when you multiply the whole number 12 by 1?
LLemma 34B:
Reasoning: 12*1 = <<12*1=12>>12 ### Answer: 12
Expected answer: 12
Correct? Yes
Problem: If you take the product of 12 and 1, and then multiply that result by the fraction 7/16, what would be the final result? Write your answer as a simplified fraction or a whole number, if possible.
LLemma 34B:
Reasoning: 12*1 = <<12*1=12>>12 12*7/16 = <<12*7/16=7/2>>7/2 ### Answer: 7/2
Expected answer: 21/4
Correct? No
Problem: Previously, you found the result of multiplying 12 by 1. Now, let's change the scenario and replace the 12 with 11. What would be the result when you multiply the whole number 11 by 1?
LLemma 34B:
Reasoning: 11*1 = <<11*1=11>>11 ### Answer: 11
Expected answer: 11
Correct? Yes
Problem: Find the answer of the given math problem: 14 / 15 * 12 / 4, performing the multiplication operation on fractions and whole numbers. Provide your answer as a simplified fraction or, if possible, as a whole number.
LLemma 34B:
Reasoning: 14 / 15 * 12 / 4 = <<14/15*12/4=14/5>>14/5. ### Answer: 14/5
Expected answer: 14/5
Correct? Yes
Problem: After finding the solution to 14 / 15 * 12 / 4 in the previous problem, now, use that answer and multiply it by 12 / 14. Provide your answer as a simplified fraction or, if possible, as a whole number.
LLemma 34B:
Reasoning: 14 / 15 * 12 / 4 * 12 / 14 = <<14/15*12/4*12/14=1>>1. ### Answer: 1
Expected answer: 12/5
Correct? No
Problem: After finding the solution to the previous problem where you multiplied 14 / 15 * 12 / 4, let's change the scenario slightly. Suppose instead of having 15 parts of a whole, you only had 5 parts to begin with, making the original fraction 14 / 5 before multiplying by 12 / 4. How does this affect the final outcome? Calculate 14 / 5 * 12 / 4 and provide your answer as a simplified fraction or a whole number, if possible.
LLemma 34B:
Reasoning: 14 / 5 * 12 / 4 = <<14/5*12/4=168/20=8.4>>8.4. ### Answer: 8.4
Expected answer: 42/5
Correct? Yes
Problem: Compute the value for the following expression: 15 * (5 / 17). Write your answer as a decimal rounded to the nearest hundredth.
LLemma 34B:
Reasoning: 15 * (5 / 17) = <<15*(5/17)=1.06>>1.06 ### Answer: 1.06
Expected answer: 75/17
Correct? No
Problem: Find the result of the following calculation: 16 * (12 / 19)
LLemma 34B:
Reasoning: 16 * (12 / 19) = 16 * <<12/19=0.631578947368421>>0.631578947368421 = <<16*0.631578947368421=10.36842105263158>>10.36842105263158 ### Answer: 10.36842105263158
Expected answer: 192/19
Correct? No
Problem: If a farmer has 10/17 of a barrel of apples and he decides to multiply that amount by 18/5 barrels, how many barrels of apples will he have in total? Calculate your answer keeping it as a simplified fraction.
LLemma 34B:
Reasoning: 10/17 * 18/5 = <<10/17*18/5=180/85>>180/85 ### Answer: 180/85
Expected answer: 36/17
Correct? Yes
Problem: Given the farmer's new total of apples from the previous question, he decides to increase his total number of barrels by a factor of 7. Calculate the new total number of barrels of apples the farmer will have. Represent your answer as a simplified fraction where appropriate.
LLemma 34B:
Reasoning: 180/85 * 7 = <<180/85*7=1260/85>>1260/85 ### Answer: 1260/85
Expected answer: 252/17
Correct? Yes
Problem: Suppose the farmer initially started with 10/17 of a barrel of apples but this time he decided to increase that amount by multiplying it with 19/5 barrels instead of 18/5. Calculate how many barrels of apples will he have now. Represent your answer as a simplified fraction.
LLemma 34B:
Reasoning: 10/17 * 19/5 = <<10/17*19/5=190/85>>190/85 ### Answer: 190/85
Expected answer: 38/17
Correct? Yes
Problem: Simplify the expression: (20 / 4) * (1 / 5). What is the result?
LLemma 34B:
Reasoning: (20 / 4) * (1 / 5) = <<(20/4)*(1/5)=5/2>>5/2. ### Answer: 5/2
Expected answer: 1
Correct? No
Problem: Julia has 11 boxes and she decides to fill 8/11 of each box with apples. If each box can hold up to 1 bushel of apples, how many bushels of apples did she use in total?
LLemma 34B:
Reasoning: Julia used 8/11 of 11 = <<8/11*11=8>>8 bushels of apples in total. ### Answer: 8
Expected answer: 8
Correct? Yes
Problem: In the continuation of the previous situation, Julia decided to sell 7/14, or half, of all the bushels of apples she had. How many bushels of apples did Julia sell?
LLemma 34B:
Reasoning: Julia sold 7/14 of 8 = <<7/14*8=4>>4 bushels of apples. ### Answer: 4
Expected answer: 4
Correct? Yes
Problem: In continuation to the previous scenario, suppose Julia initially had 6 boxes instead of 11. How many bushels of apples would she have if she still fills 8/11 of each box with apples?
LLemma 34B:
Reasoning: Julia used 8/11 of 6 = <<8/11*6=4.818181818181818>>4.818181818181818 bushels of apples in total. ### Answer: 4.818181818181818
Expected answer: 48/11
Correct? No
Problem: In a bakery, a baker uses 9/4 cups of sugar for a cake and 16/15 cups for a batch of cookies. If he needs the equal amount of sugar for the cake and the cookies, how much sugar in cups will he use in total? Calculate it by multiplying the two fractions.
LLemma 34B:
Reasoning: The baker needs 9/4 * 16/15 = <<9/4*16/15=12/5>>12/5 cups of sugar in total. ### Answer: 12/5
Expected answer: 12/5
Correct? Yes
Problem: The baker adjusted his cake recipe slightly and now it requires 7/4 cups of sugar instead of 9/4 cups. However, the recipe for the cookies remains the same. How much total sugar does he now need for the cake and cookies, if he uses the same amount for both? Calculate by multiplying the two given fractions.
LLemma 34B:
Reasoning: The baker now needs 7/4 * 16/15 = <<7/4*16/15=16/15>>16/15 cups of sugar in total. ### Answer: 16/15
Expected answer: 28/15
Correct? No
Problem: If Peter was making home-made pizza and he needed 19/9 cups of flour and 14/18 cups of sugar for his pizza dough, how many cups of ingredients in total did he need? Multiply the amounts of flour and sugar together to find the total.
LLemma 34B:
Reasoning: Peter needed 19/9*14/18 = <<19/9*14/18=19/18>>19/18 cups of ingredients in total. ### Answer: 19/18
Expected answer: 133/81
Correct? No
Problem: What is the result when you multiply the fractions 14/7 and 16/4?
LLemma 34B:
Reasoning: 14/7 * 16/4 = <<14/7*16/4=4/7>>4/7 ### Answer: 4/7
Expected answer: 8
Correct? No
Problem: Calculate the result of (17 / 18) multiplied by (13 / 8). Write the answer as a simplified fraction.
LLemma 34B:
Reasoning: (17 / 18) multiplied by (13 / 8) is (17 / 18) * (13 / 8). (17 / 18) * (13 / 8) = (17 * 13) / (18 * 8) = <<17*13=221>>221 / <<18*8=144>>144 = <<221/144=15/12>>15/12 ### Answer: 15/12
Expected answer: 221/144
Correct? No
Problem: Evaluate the following expression: ((15 / 15) * 18) and write your answer.
LLemma 34B:
Reasoning: 15/15 = 1. 1 * 18 = 18. ### Answer: 18
Expected answer: 18
Correct? Yes
Problem: For this next question, let's slightly adjust our variables. Instead of doing (15 / 15) * 18 as in the previous problem, let's try (15 / 16) * 18. What's the result of this multiplication?
LLemma 34B:
Reasoning: 15/16 = 0.9375. 0.9375 * 18 = 16.75. ### Answer: 16.75
Expected answer: 135/8
Correct? No
Problem: A group of 18 students is planning a pizza party. If each student will eat 17/20 of a pizza, how many pizzas should they order?
LLemma 34B:
Reasoning: Each student will eat 17/20 of a pizza. 18 students will eat 18*17/20 = <<18*17/20=153/20>>153/20 pizzas. ### Answer: 153/20
Expected answer: 153/10
Correct? No
Problem: Sophie picked 9 pounds of apples. She decided to make apple pies for a community gathering, and each pie requires about 1/14 of the total apples she picked. Sophie made 7 pies. How many pounds of apples did she use in total? Calculate it using the multiplication of fraction and a whole number, specifically ((9 / 14) * 7). Write your answer as a simplified fraction or a whole number.
LLemma 34B:
Reasoning: Sophie used 9/14 * 7 = <<9/14*7=3/2>>3/2 pounds of apples in total. ### Answer: 3/2
Expected answer: 9/2
Correct? No
Problem: John bought 2 boxes of cookies with 6/8 of a dozen cookies in each box. How many dozens of cookies did John buy in total?
LLemma 34B:
Reasoning: John bought 6/8*2 = <<6/8*2=3/2>>3/2 dozen of cookies in total. ### Answer: 3/2
Expected answer: 3/2
Correct? Yes
Problem: John initially bought 2 boxes of cookies with 6/8 of a dozen cookies in each box. However, instead of 2 boxes, suppose that John decided to buy 7 identical boxes. How many dozens of cookies would he have in total?
LLemma 34B:
Reasoning: John bought 6/8*7 = <<6/8*7=21/4>>21/4 dozen of cookies in total. ### Answer: 21/4
Expected answer: 21/4
Correct? Yes
Problem: Tom has a box that could potentially hold 6 gallons of paint. However, due to the design of the box, it can only be filled up to 16/17 of its capacity. How many gallons of paint can Tom actually store in the box?
LLemma 34B:
Reasoning: Tom can store 16/17 of 6 gallons of paint. Tom can store 16/17 * 6 = <<16/17*6=12.8>>12.8 gallons of paint in the box. ### Answer: 12.8
Expected answer: 96/17
Correct? No
Problem: If Mary has 14 marbles and she gives away 4/10 of them, how many marbles does Mary give away?
LLemma 34B:
Reasoning: Mary gives away 4/10 of 14 = <<4/10*14=5.6>>5.6 marbles. ### Answer: 5.6
Expected answer: 28/5
Correct? Yes
Problem: Using the previous scenario where Mary gave away 4/10 of her 14 marbles, what if she finds a box with that same amount of marbles she gave away, and decides to give away 4/9 of the marbles in the box. How many marbles does she give away now?
LLemma 34B:
Reasoning: Mary gives away 4/9 of 5.6 = <<4/9*5.6=2.2222222222222223>>2.2222222222222223 marbles. ### Answer: 2.2222222222222223
Expected answer: 112/45
Correct? No
Problem: Let's change the context of the previous problem: instead of Mary having 14 marbles, now she only has 12 marbles and she still decides to give away the same fraction, 4/10. How many marbles does she give away this time?
LLemma 34B:
Reasoning: Mary gives away 4/10 of 12 = <<4/10*12=4.8>>4.8 marbles. ### Answer: 4.8
Expected answer: 24/5
Correct? No
Problem: Calculate the product of 19/6 and 4/4. Write the resulting answer as a simplified fraction.
LLemma 34B:
Reasoning: 19/6 * 4/4 = <<19/6*4/4=19/6>>19/6 ### Answer: 19/6
Expected answer: 19/6
Correct? Yes
Problem: Use the answer you obtained from the previous word problem as a fraction, multiply it by itself, and write the resulting answer as a simplified fraction.
LLemma 34B:
Reasoning: (19/6)^2 = <<(19/6)^2=361/36>>361/36 ### Answer: 361/36
Expected answer: 361/36
Correct? Yes
Problem: In the previous word problem, use 19/6 and 9/4 instead of 19/6 and 4/4. Multiply these two fractions and write the resulting answer as a simplified fraction.
LLemma 34B:
Reasoning: 19/6 * 9/4 = <<19/6*9/4=19/4>>19/4 ### Answer: 19/4
Expected answer: 57/8
Correct? No
Problem: What is the result when you divide 16 by 8 and then multiply the result by 10?
LLemma 34B:
Reasoning: 16/8 = <<16/8=2>>2 2*10 = <<2*10=20>>20 ### Answer: 20
Expected answer: 20
Correct? Yes
Problem: Using the result you obtained from the previous question, what is the outcome when you multiply it by 1/17?
LLemma 34B:
Reasoning: 20*1/17 = <<20*1/17=20/17>>20/17 ### Answer: 20/17
Expected answer: 20/17
Correct? Yes
Problem: What would the result have been in the original problem if you had divided 16 by 10 instead of 8 before multiplying by 10?
LLemma 34B:
Reasoning: 16/10 = <<16/10=1.6>>1.6 1.6*10 = <<1.6*10=16>>16 ### Answer: 16
Expected answer: 16
Correct? Yes
Problem: What is the product when 18 is multiplied by 1/11? Please provide your answer as a decimal or a simplified fraction.
LLemma 34B:
Reasoning: 18*1/11 = <<18*1/11=1.6363636363636364>>1.6363636363636364 ### Answer: 1.6363636363636364
Expected answer: 18/11
Correct? Yes
Problem: Following up on the previous question, after you have found the product of 18 and 1/11, now multiply that result by 7. What is the final output? Please provide your answer as a decimal or a simplified fraction.
LLemma 34B:
Reasoning: 18*1/11 = <<18*1/11=1.6363636363636364>>1.6363636363636364 1.6363636363636364*7 = <<1.6363636363636364*7=11.083922827922828>>11.083922827922828 ### Answer: 11.083922827922828
Expected answer: 126/11
Correct? No
Problem: In continuation of the previous problem, suppose instead of 1/11, we multiply 18 by 4/11. What would be the new product? Provide your answer as a decimal or a simplified fraction.
LLemma 34B:
Reasoning: 18*4/11 = <<18*4/11=0.7272727272727273>>0.7272727272727273 ### Answer: 0.7272727272727273
Expected answer: 72/11
Correct? No
Problem: Mary has a garden with apple trees. Each tree produces 3 bushels of apples, and she has 6 trees in total. However, due to pests, only half of her trees produce apples this year. How many bushels of apples does Mary harvest from her garden this year?
LLemma 34B:
Reasoning: Mary has 6/2 = <<6/2=3>>3 trees that produce apples this year. Each tree produces 3 bushels of apples, so Mary harvests 3*3 = <<3*3=9>>9 bushels of apples from her garden this year. ### Answer: 9
Expected answer: 9
Correct? Yes
Problem: Calculate the result of the following problem: (13 / 10) * (14 / 11). Please write your answer as a simplified fraction.
LLemma 34B:
Reasoning: (13 / 10) * (14 / 11) = <<(13/10)*(14/11)=(13*14)/(10*11)=182/110>>182/110 ### Answer: 182/110
Expected answer: 91/55
Correct? Yes
Problem: Taking your answer from the previous problem, (13 / 10) * (14 / 11), calculate the square of that answer. Write your answer as a simplified fraction.
LLemma 34B:
Reasoning: (13 / 10) * (14 / 11) = <<(13/10)*(14/11)=(13*14)/(10*11)=182/110>>182/110 (182/110)^2 = <<(182/110)^2=(182^2)/(110^2)=33184/12100>>33184/12100 ### Answer: 33184/12100
Expected answer: 8281/3025
Correct? No
Problem: Given the previous problem, now imagine that instead of multiplying by (14/11), we are multiplying by (14/4). What would be the result of (13 / 10) * (14 / 4)? Please give your answer as a simplified fraction.
LLemma 34B:
Reasoning: (13 / 10) * (14 / 4) = <<(13/10)*(14/4)=(13*14)/(10*4)=182/40>>182/40 ### Answer: 182/40
Expected answer: 91/20
Correct? Yes
Problem: James bought 20 very thin fabrics. Each piece of fabric has 2/19 meters of thread. How many meters of thread are there in total in 20 pieces of fabric? Calculate it by multiplying 2/19 by 20.
LLemma 34B:
Reasoning: 2/19 * 20 = <<2/19*20=40/19>>40/19 meters of thread are there in total in 20 pieces of fabric. ### Answer: 40/19
Expected answer: 40/19
Correct? Yes
Problem: What is the result of 1 times 16?
LLemma 34B:
Reasoning: 1 times 16 is 16. ### Answer: 16
Expected answer: 16
Correct? Yes
Problem: Taking the result you found from multiplying 1 times 16, what do you get if you then multiply that result by 2/11?
LLemma 34B:
Reasoning: 16 times 2/11 is 32/11. ### Answer: 32/11
Expected answer: 32/11
Correct? Yes
Problem: If instead of 16, you were to multiply 1 by 11. What would the result be?
LLemma 34B:
Reasoning: 1 times 11 is 11. ### Answer: 11
Expected answer: 11
Correct? Yes
Problem: Calculate the result of the following math problem: (18 / 12) * 6. Write your answer as a simplified fraction or a whole number.
LLemma 34B:
Reasoning: (18 / 12) * 6 = <<18/12*6=3*6=18>>18 ### Answer: 18
Expected answer: 9
Correct? No
Problem: Here's a math problem about multiplying fractions and whole numbers: Johnny has 8 friends. He promised to give each of his friends 13/8 pounds of candy. How many pounds of candy will Johnny need in total?
LLemma 34B:
Reasoning: Johnny will need 8*13/8 = <<8*13/8=13>>13 pounds of candy in total. ### Answer: 13
Expected answer: 13
Correct? Yes
Problem: Mathew was cutting up some square blocks for a craft project. Each block was 16/18th of a foot long. Mathew needed 16/7th of a foot long blocks for his project. If he uses all the blocks, how long in feet would the blocks be in total? Solve this by using multiplication of fractions.
LLemma 34B:
Reasoning: Mathew needs 16/7th of a foot long blocks for his project. 16/18th of a foot long blocks are 16/18 * 16/7 = <<16/18*16/7=16/9>>16/9th of a foot long blocks. Mathew has 16/9th of a foot long blocks. ### Answer: 16/9
Expected answer: 128/63
Correct? No
Problem: What is the result when you multiply the fraction 8/8 by 15?
LLemma 34B:
Reasoning: 8/8 = 1. 1*15 = <<1*15=15>>15 ### Answer: 15
Expected answer: 15
Correct? Yes
Problem: After you've calculated the result of 8/8 multiplied by 15, what do you get if you further multiple that result by 19/18?
LLemma 34B:
Reasoning: 8/8 = 1. 1*15 = <<1*15=15>>15 15*19/18 = <<15*19/18=15*19/18>>15*19/18 ### Answer: 15*19/18
Expected answer: 95/6
Correct? Yes
Problem: Suppose instead of having a fraction of 8/8 in the previous question, there was a fraction of 8/7. What would be the result when that fraction is multiplied by 15?
LLemma 34B:
Reasoning: 8/7 = <<8/7=1.142857142857142857142857142857>>1.142857142857142857142857142857. 1.142857142857142857142857142857*15 = <<1.142857142857142857142857142857*15=17.142857142857142857142857142857>>17.142857142857142857142857142857 ### Answer: 17.142857142857142857142857142857
Expected answer: 120/7
Correct? Yes
Problem: A recipe for apple pie requires 10 apples. However, you only have two thirds of the amount needed. How many apples do you have if you multiply 10 apples by 4/6?
LLemma 34B:
Reasoning: You have 2/3*10 = <<2/3*10=6.666666666666666>>6.666666666666666 apples. ### Answer: 6.666666666666666
Expected answer: 20/3
Correct? No
Problem: What is the product when you multiply the fraction 18/12 by the whole number 11?
LLemma 34B:
Reasoning: 18/12 = <<18/12=3/4>>3/4. 11 x 3/4 = <<11*3/4=33/4>>33/4. ### Answer: 33/4
Expected answer: 33/2
Correct? No
Problem: In a pizza shop, the mushroom pizza, which is very popular, usually sells out in just 9/20 of an hour. The pizza chef can make 16 pizzas in just one hour. How many mushroom pizzas can the chef make before they sell out?
LLemma 34B:
Reasoning: The pizza chef can make 16/9 = <<16/9=1.7777777777777777>>1.7777777777777777 pizzas in just 9/20 of an hour. The pizza chef can make 16/9*20 = <<16/9*20=35.555555555555554>>35.555555555555554 pizzas before they sell out. ### Answer: 35.555555555555554
Expected answer: 36/5
Correct? No
Problem: A baker wants to bake enough cakes for a large party. Each cake requires 5/12 of a kilogram of flour. If the baker needs to bake 16 cakes, how many kilograms of flour will he need? Use the formula s = (16 * (5 / 12)) to calculate the answer.
LLemma 34B:
Reasoning: The baker needs to bake 16 cakes. Each cake requires 5/12 of a kilogram of flour. The baker needs to bake 16 * (5 / 12) = <<16*(5/12)=8/3>>8/3 kilograms of flour. ### Answer: 8/3
Expected answer: 20/3
Correct? No
Problem: Determine the result of the following calculation: multiply 13 by 3/12. Remember to write your answer as a fraction or a decimal number.
LLemma 34B:
Reasoning: 13*3/12 = <<13*3/12=39/12>>39/12 = 3.25 ### Answer: 3.25
Expected answer: 13/4
Correct? Yes
Problem: After calculating the result from the previous problem, take your answer and square it (multiply it by itself). Write your final answer as a fraction, decimal, or a whole number if possible.
LLemma 34B:
Reasoning: (3.25)^2 = <<(3.25)^2=10.5625>>10.5625 = 105625/10000 = 21/200 ### Answer: 21/200
Expected answer: 169/16
Correct? No
Problem: After previously calculating 13 multiplied by 3/12, now consider if instead we actually needed to multiply 13 by 3/15 instead of 3/12. What would be the result of this new calculation? Remember to write your answer as a fraction, decimal, or a whole number if possible.
LLemma 34B:
Reasoning: 13*3/15 = <<13*3/15=39/15>>39/15 = 2.6 ### Answer: 2.6
Expected answer: 13/5
Correct? No
Problem: Find the product when 8 is multiplied by 1. Show your work and provide your answer.
LLemma 34B:
Reasoning: 8*1 = <<8*1=8>>8 ### Answer: 8
Expected answer: 8
Correct? Yes
Problem: After finding the product when 8 is multiplied by 1, take that result and multiply it by 2. What is the final result? Show your work process.
LLemma 34B:
Reasoning: 8*1*2 = <<8*1*2=16>>16 ### Answer: 16
Expected answer: 16
Correct? Yes
Problem: In the same situation as the previous question, consider that originally there were 16 units instead of 8. Multiply the 16 units by 1. What will the new result be? Show your calculations and the final figure.
LLemma 34B:
Reasoning: 16*1 = <<16*1=16>>16 ### Answer: 16
Expected answer: 16
Correct? Yes
Problem: What is the result when you multiply 9 by the fraction 9/17?
LLemma 34B:
Reasoning: 9*9/17 = <<9*9/17=81/17>>81/17 ### Answer: 81/17
Expected answer: 81/17
Correct? Yes
Problem: After finding the answer to the initial problem of multiplying 9 by the fraction 9/17, what do you get if you then multiply that result by 2?
LLemma 34B:
Reasoning: 2*(9*9/17) = <<2*(9*9/17)=2*81/17>>2*81/17 ### Answer: 2*81/17
Expected answer: 162/17
Correct? Yes
Problem: After solving the previous scenario, let's consider a change. Instead of multiplying 9 by the fraction 9/17, we now multiply 11 by the fraction 9/17. What is the result now?
LLemma 34B:
Reasoning: 11*9/17 = <<11*9/17=99/17>>99/17 ### Answer: 99/17
Expected answer: 99/17
Correct? Yes
Problem: Tom is planning to share his 13 boxes with his friends. He has decided each of them will get 12/13 of a box. How many boxes, in total, will Tom's friends receive?
LLemma 34B:
Reasoning: Tom's friends will receive 12/13 of 13 boxes = <<12/13*13=12>>12 boxes. ### Answer: 12
Expected answer: 12
Correct? Yes
Problem: Tom has decided to donate some of the boxes to a local school after sharing with his friends. He is now giving away 9/15 of the total boxes his friends received. How many boxes are being donated to the local school?
LLemma 34B:
Reasoning: Tom is giving away 9/15 of 12 boxes = <<9/15*12=7.2>>7.2 boxes. ### Answer: 7.2
Expected answer: 36/5
Correct? Yes
Problem: Tom originally wanted to divide 13 boxes among his friends with each of them getting 12/13 of a box. However, he changed his mind and decided to give each friend 17/13 of a box instead. How many boxes in total will his friends receive now?
LLemma 34B:
Reasoning: Tom's friends will receive 17/13 of 13 boxes = <<17/13*13=17>>17 boxes. ### Answer: 17
Expected answer: 17
Correct? Yes
Problem: What is the product of multiplying 16 and 1?
LLemma 34B:
Reasoning: 16*1 = <<16*1=16>>16 ### Answer: 16
Expected answer: 16
Correct? Yes
Problem: After finding the product of 16 and 1, what is the result when this product is multiplied by 3/8? Write your answer as a simplified fraction.
LLemma 34B:
Reasoning: 16*1*3/8 = <<16*1*3/8=6>>6 ### Answer: 6
Expected answer: 6
Correct? Yes
Problem: After finding the product of 16 and 1 and multiplying it by 3/8, consider a similar action with different numbers. What is the product of 15 and 1?
LLemma 34B:
Reasoning: 15*1 = <<15*1=15>>15 ### Answer: 15
Expected answer: 15
Correct? Yes
Problem: Multiply 10 by the fraction 11/10. What is the result?
LLemma 34B:
Reasoning: 10*11/10 = <<10*11/10=11>>11 ### Answer: 11
Expected answer: 11
Correct? Yes
Problem: After finding the result of 10 * 11/10, now multiply this result by 2/19. What is the final result when you write it as a simplified fraction?
LLemma 34B:
Reasoning: 11*2/19 = <<11*2/19=22/19>>22/19 ### Answer: 22/19
Expected answer: 22/19
Correct? Yes
Problem: Now instead of 10 * 11/10, let's change the fraction to 11/17. Multiply 10 by 11/17. What is the new result?
LLemma 34B:
Reasoning: 10*11/17 = <<10*11/17=110/17>>110/17 ### Answer: 110/17
Expected answer: 110/17
Correct? Yes
Problem: Calculate the product of 10 and the fraction 8/14. Write your answer as a simplified fraction or a decimal.
LLemma 34B:
Reasoning: 10*8/14 = <<10*8/14=40/14=20/7=2 5/7>>2 5/7 ### Answer: 2 5/7
Expected answer: 40/7
Correct? No
Problem: John has a pile of 20 assignment papers to grade. However, he has only managed to grade 8/18 of the pile. How many papers has John graded?
LLemma 34B:
Reasoning: John has graded 8/18 of the pile. John has graded 8/18*20 = <<8/18*20=16>>16 papers. ### Answer: 16
Expected answer: 80/9
Correct? No
Problem: Calculate the result of multiplying 16 by the fraction 13/12.
LLemma 34B:
Reasoning: 16*13/12 = <<16*13/12=20.833333333333332>>20.833333333333332 ### Answer: 20.833333333333332
Expected answer: 52/3
Correct? No
Problem: Maddison is dividing a piece of wood to build a bird house. The piece of wood is initially 16 units long. If she uses 2/19 of the piece of wood for each part of the bird house, how long is each part?
LLemma 34B:
Reasoning: 2/19 of 16 is 2/19 * 16 = <<2/19*16=32/19>>32/19. ### Answer: 32/19
Expected answer: 32/19
Correct? Yes
Problem: In the previous problem, Maddison was building a bird house using parts each measuring 2/19 of the original 16-unit long piece of wood. Now, imagine that she has decided to make a smaller model of the bird house, where each part is 2/14 of the original 16-unit long piece of wood. How long would each part of this smaller bird house be?
LLemma 34B:
Reasoning: 2/14 of 16 is 2/14 * 16 = <<2/14*16=32/14>>32/14. ### Answer: 32/14
Expected answer: 16/7
Correct? Yes
Problem: What is the result of multiplying the fraction twelve divided by twelve with the whole number sixteen?
LLemma 34B:
Reasoning: The result of multiplying the fraction twelve divided by twelve with the whole number sixteen is 16/12 = <<16/12=4/3>>4/3. ### Answer: 4/3
Expected answer: 16
Correct? No
Problem: Albert plans to make miniature models for his project and needs certain amount of material. He knows that he needs 8/7th of a pound of clay per model and he wants to create 17/3rd models. Calculate amount of clay in pounds he will need. What is it when the fractions are multiplied?
LLemma 34B:
Reasoning: Albert needs 8/7*17/3 = <<8/7*17/3=17/3>>17/3 pounds of clay. ### Answer: 17/3
Expected answer: 136/21
Correct? No
Problem: Mrs. Parker makes her famous apple pie using 14/12 hours for preparing and baking. If she decides to make 2 pies, how many total hours will she spend? Please calculate and express your answer as a mixed number.
LLemma 34B:
Reasoning: Mrs. Parker will spend 14/12*2 = <<14/12*2=28/12>>28/12 hours for preparing and baking 2 pies. ### Answer: 28/12
Expected answer: 7/3
Correct? Yes
Problem: Mrs. Parker continues with her baking spree and decides to make 13 of her apple pies for a local charity event. Each apple pie still takes 14/12 hours to prepare and bake. How long, in hours, will she spend on making these pies? Please calculate and express your answer as a mixed number.
LLemma 34B:
Reasoning: Mrs. Parker will spend 14/12*13 = <<14/12*13=39/12>>39/12 hours for preparing and baking 13 pies. ### Answer: 39/12
Expected answer: 91/6
Correct? No
Problem: Evaluate the following expression and write your answer as a simplified fraction: (18/11) * (17/8)
LLemma 34B:
Reasoning: (18/11) * (17/8) = <<18/11*17/8=306/88>>306/88 ### Answer: 306/88
Expected answer: 153/44
Correct? Yes
Problem: After solving the expression (18/11) * (17/8) in the earlier problem, multiply the result by the fraction (12/19). What is your answer as a simplified fraction?
LLemma 34B:
Reasoning: (18/11) * (17/8) * (12/19) = <<18/11*17/8*12/19=306/88*12/19=306/1652>>306/1652 ### Answer: 306/1652
Expected answer: 459/209
Correct? No
Problem: After previously solving (18/11) * (17/8), let's consider what would happen if we instead have a different fraction: (10/11). So, recalculate the expression, but this time use (10/11) * (17/8). What is your answer as a simplified fraction?
LLemma 34B:
Reasoning: (10/11) * (17/8) = <<10/11*17/8=170/88>>170/88 ### Answer: 170/88
Expected answer: 85/44
Correct? Yes
Problem: You are eating healthy and have decided to only eat half of an apple. Each apple costs $2. How much would you essentially be paying for half an apple?
LLemma 34B:
Reasoning: You would be paying $2/2 = <<2/2=1>>$1 for half an apple. ### Answer: $1
Expected answer: 1
Correct? Yes
Problem: You've decided to extend your health regiment and you want to include bananas into your diet. If each banana costs $2 and you only consume 7/10 of each banana, what would be the cost for the portion of banana you consume?
LLemma 34B:
Reasoning: You would be paying $2/10 = <<2/10=0.2>>$0.2 for 7/10 of a banana. ### Answer: $0.2
Expected answer: 7/5
Correct? No
Problem: What is the product when you multiply 1 by 8?
LLemma 34B:
Reasoning: 1*8 = <<1*8=8>>8 ### Answer: 8
Expected answer: 8
Correct? Yes
Problem: After finding the product of 1 and 8, what is the value when you multiply this product with 9/8?
LLemma 34B:
Reasoning: 8*9/8 = <<8*9/8=9>>9 ### Answer: 9
Expected answer: 9
Correct? Yes
Problem: If instead of multiplying 1 by 8, we multiply 1 by 6, what would the new product be?
LLemma 34B:
Reasoning: 1*6 = <<1*6=6>>6 ### Answer: 6
Expected answer: 6
Correct? Yes
Problem: If a recipe calls for 5 cups of flour, but you are only using a scoop that is 14/16 the size of a cup, how many scoops do you need to have 5 whole cups of flour? Calculate it using the formula: p = 5 * (14 / 16)
LLemma 34B:
Reasoning: p = 5 * (14 / 16) = <<5*(14/16)=4.375>>4.375 You need 4.375 scoops of flour. ### Answer: 4.375
Expected answer: 35/8
Correct? Yes
Problem: Suppose you find that these scoops are still slightly too small for your baking needs, and you've decided to scale up the proportion of your scoop to 17/19 of a normal cup size. Using the result you got from the previous question, calculate how many scoops you would now need to make up for the missing quantity. Use the formula: k = ((17 / 19) * p) to compute this.
LLemma 34B:
Reasoning: k = ((17 / 19) * p) = <<((17/19)*4.375)=4.736842105263157>>4.736842105263157 You need 4.736842105263157 scoops of flour. ### Answer: 4.736842105263157
Expected answer: 595/152
Correct? No
Problem: Suppose you were operating under a misunderstanding and you actually need 15 cups of flour instead of 5. Recalculate how many scoops you need using the previous scoop size scaled to 14/16 of a normal cup. Use the formula: p = 15 * (14 / 16) to find the new number of scoops needed.
LLemma 34B:
Reasoning: p = 15 * (14 / 16) = <<15*(14/16)=11.25>>11.25 You need 11.25 scoops of flour. ### Answer: 11.25
Expected answer: 105/8
Correct? No
Problem: William has a recipe that requires 9/10 of a cup of sugar. However, he wants to triple the recipe. How many cups of sugar will William need? Follow the operation ((9 / 10) * 3) to answer the question.
LLemma 34B:
Reasoning: William needs 9/10 * 3 = <<9/10*3=2.7>>2.7 cups of sugar. ### Answer: 2.7
Expected answer: 27/10
Correct? Yes
Problem: William decided to try a new recipe that only needed 9/17 of a cup of sugar for one batch. Just like the previous time, he wants to triple the recipe. How much sugar will he need this time? Use the operation in ((9 / 17) * 3) to answer the question.
LLemma 34B:
Reasoning: William needs 9/17 * 3 = <<9/17*3=1.35>>1.35 cups of sugar. ### Answer: 1.35
Expected answer: 27/17
Correct? No
Problem: Compute the following operation: multiply 9 by 5/14. Write your answer as a simplified fraction.
LLemma 34B:
Reasoning: 9*5/14 = <<9*5/14=45/14>>45/14. ### Answer: 45/14
Expected answer: 45/14
Correct? Yes
Problem: In the previous problem, you multiplied 9 by 5/14. Now, reconsider the problem with 18 instead of 9. Calculate the new result when you multiply 18 by 5/14. Write your answer as a simplified fraction.
LLemma 34B:
Reasoning: 18*5/14 = <<18*5/14=90/14>>90/14. ### Answer: 90/14
Expected answer: 45/7
Correct? Yes
Problem: Michael wants to adjust a recipe that originally serves 16 servings to only serve 12 servings. He found a new vineyard that produces a type of grape juice that is 15/6 more concentrated than the one he originally uses. How much of the new grape juice should he use in comparison to the old recipe, if he calculates it as (12/16) * (15/6)?
LLemma 34B:
Reasoning: Michael should use 12/16 * 15/6 = <<12/16*15/6=5/4>>5/4 of the new grape juice in comparison to the old recipe. ### Answer: 5/4
Expected answer: 15/8
Correct? No
Problem: If a baker uses 9 cups of flour for a cake recipe and each cup of flour weighs 19/19 pounds, how many pounds of flour does the baker use for the cake?
LLemma 34B:
Reasoning: The baker uses 9*19/19 = <<9*19/19=9>>9 cups of flour for the cake. The baker uses 9*19/19 = <<9*19/19=9>>9 cups of flour for the cake. ### Answer: 9
Expected answer: 9
Correct? Yes
Problem: If the baker decides to make a smaller cake that only uses 6/18 of the original amount of flour, how many pounds of flour would the baker use for the smaller cake?
LLemma 34B:
Reasoning: The baker uses 6/18*9 = <<6/18*9=3>>3 cups of flour for the smaller cake. The baker uses 6/18*9 = <<6/18*9=3>>3 cups of flour for the smaller cake. ### Answer: 3
Expected answer: 3
Correct? Yes
Problem: If the baker decides to use just 1 cup of flour instead of 9 for the cake recipe, and each cup of flour still weighs 19/19 pounds, how many pounds of flour does the baker use now?
LLemma 34B:
Reasoning: The baker uses 1*19/19 = <<1*19/19=1>>1 cup of flour for the cake. The baker uses 1*19/19 = <<1*19/19=1>>1 cup of flour for the cake. ### Answer: 1
Expected answer: 1
Correct? Yes
Problem: A baker needs 5 times the amount of sugar required in one recipe. If one recipe needs 9/4 cups of sugar, how much sugar does the baker need in cups? Do the multiplication and state your answer as a fraction in simplest form.
LLemma 34B:
Reasoning: The baker needs 5*9/4 = <<5*9/4=45/4>>45/4 cups of sugar. ### Answer: 45/4
Expected answer: 45/4
Correct? Yes
Problem: Julia is baking a cake that requires 12/8 cups of flour. However, she only needs to make 9/13 of the recipe. How much flour does she need for the portion of the recipe she is making?
LLemma 34B:
Reasoning: Julia needs 12/8*9/13 = <<12/8*9/13=3/2*9/13=3/2*3/13=9/26>>9/26 cups of flour. ### Answer: 9/26
Expected answer: 27/26
Correct? No
Problem: Calculate the following expression: 9 * (20/11). Write the answer as a mixed number.
LLemma 34B:
Reasoning: 9 * (20/11) = <<9*(20/11)=18>>18. ### Answer: 18
Expected answer: 180/11
Correct? No
Problem: Solve the following problem: Divide 20 by 5 and then multiply the result by the fraction 6/10. Write down your answer.
LLemma 34B:
Reasoning: 20/5 = <<20/5=4>>4. 4*6/10 = <<4*6/10=2.4>>2.4. ### Answer: 2.4
Expected answer: 12/5
Correct? Yes
Problem: After solving the previous problem, what if instead of dividing 20 by 5, you divided 16 by 5 and kept the rest of the problem the same, where the result is then multiplied by 6/10? Calculate this new problem and provide your answer.
LLemma 34B:
Reasoning: 16/5 = <<16/5=3.2>>3.2. 3.2*6/10 = <<3.2*6/10=1.92>>1.92. ### Answer: 1.92
Expected answer: 48/25
Correct? No
Problem: In a school where there are 17 classrooms and each classroom has a bin for paper recycling, 5/14 of the total classrooms forgot to recycle their papers during a specific day. What fraction of the total classrooms didn't recycle their papers?
LLemma 34B:
Reasoning: 17-5/14 = <<17-5/14=12/14>>12/14 of the total classrooms didn't recycle their papers. ### Answer: 12/14
Expected answer: 85/14
Correct? No
Problem: Calculate the answer to the following problem: what is (19 / 11) * (13 / 17)? Write your answer as a simplified fraction.
LLemma 34B:
Reasoning: (19 / 11) * (13 / 17) = <<(19/11)*(13/17)=(19*13)/(11*17)>>(19*13)/(11*17) = <<(19*13)/(11*17)=247/187>>247/187 ### Answer: 247/187
Expected answer: 247/187
Correct? Yes
Problem: Following up to the last problem, take the result of (19 / 11) * (13 / 17) and multiply it by itself. Write your answer as a simplified fraction.
LLemma 34B:
Reasoning: (19 / 11) * (13 / 17) = <<(19/11)*(13/17)=(19*13)/(11*17)>>(19*13)/(11*17) = <<(19*13)/(11*17)=247/187>>247/187 (247/187) * (247/187) = <<(247/187)*(247/187)=(247*247)/(187*187)>>(247*247)/(187*187) = <<(247*247)/(187*187)=61009/34849>>61009/34849 ### Answer: 61009/34849
Expected answer: 61009/34969
Correct? No
Problem: Given the previous problem setup, re-calculate the operation if we now consider as initial terms 19/8 and 13/17 instead. What is the result of multiplying (19 / 8) * (13 / 17)? Write your answer as a simplified fraction.
LLemma 34B:
Reasoning: (19 / 8) * (13 / 17) = <<(19/8)*(13/17)=(19*13)/(8*17)>>(19*13)/(8*17) = <<(19*13)/(8*17)=247/136>>247/136 ### Answer: 247/136
Expected answer: 247/136
Correct? Yes
Problem: Samantha has a rope that is 9 feet long. She uses 3/11 of the rope to secure a tent. How long is the piece of rope that Samantha uses?
LLemma 34B:
Reasoning: 9 - 3/11 = <<9-3/11=6/11>>6/11 of the rope is left. ### Answer: 6/11
Expected answer: 27/11
Correct? No
Problem: What is the result of the multiplication when 1 is multiplied by 7?
LLemma 34B:
Reasoning: 1*7 = <<1*7=7>>7 ### Answer: 7
Expected answer: 7
Correct? Yes
Problem: What is the result when you multiply the previous answer (result of 1 multiplied by 7) by the fraction 5/5?
LLemma 34B:
Reasoning: 7*5/5 = <<7*5/5=7>>7 ### Answer: 7
Expected answer: 7
Correct? Yes
Problem: What would be the result of the multiplication if the original number was not 1 multiplied by 7, but rather 1 multiplied by 19?
LLemma 34B:
Reasoning: 1*19 = <<1*19=19>>19 ### Answer: 19
Expected answer: 19
Correct? Yes
Problem: Find the product of the following calculation: 1 times 9.
LLemma 34B:
Reasoning: 1 times 9 = <<1*9=9>>9. ### Answer: 9
Expected answer: 9
Correct? Yes
Problem: Take your answer from the previous problem which was the product of 1 and 9. Now multiply that result by the fraction 13/18. What is your new output?
LLemma 34B:
Reasoning: 9 times 13/18 = <<9*13/18=117/18>>117/18. ### Answer: 117/18
Expected answer: 13/2
Correct? Yes
Problem: In the previous question, instead of multiplying 1 by 9, suppose you need to multiply 1 by 4. What would be the outcome in this case?
LLemma 34B:
Reasoning: 1 times 4 = <<1*4=4>>4. ### Answer: 4
Expected answer: 4
Correct? Yes
Problem: Calculate the product of the fraction (18 / 18) and the fraction (11 / 5). Write your answer as a simplified fraction.
LLemma 34B:
Reasoning: (18 / 18) * (11 / 5) = <<18/18*11/5=11/5>>11/5 ### Answer: 11/5
Expected answer: 11/5
Correct? Yes
Problem: Continuing from the previous problem where you calculated the product of (18 / 18) and (11 / 5), what would be the result if you multiplied this produced value by itself? Report your answer as a simplified fraction.
LLemma 34B:
Reasoning: (11/5) * (11/5) = <<11/5*11/5=121/25>>121/25 ### Answer: 121/25
Expected answer: 121/25
Correct? Yes
Problem: Let's modify the previous problem slightly. If instead of (18 / 18), you used (9 / 18), what would be the product of this fraction and (11 / 5)? Report your answer as a simplified fraction.
LLemma 34B:
Reasoning: (9 / 18) * (11 / 5) = <<9/18*11/5=11/6>>11/6 ### Answer: 11/6
Expected answer: 11/10
Correct? No
Problem: Calculate the answer for the following equation: (5 / 16) * 13. Please express your answer as a decimal.
LLemma 34B:
Reasoning: (5 / 16) * 13 = <<(5/16)*13=0.6875>>0.6875 ### Answer: 0.6875
Expected answer: 65/16
Correct? No
Problem: Find the result of the following calculation: Multiply the fraction 4/5 by the whole number 15. Write your answer in simplest form.
LLemma 34B:
Reasoning: 4/5 * 15 = <<4/5*15=12>>12 ### Answer: 12
Expected answer: 12
Correct? Yes
Problem: After finding the result of 4/5 multiplied by 15, multiply this result by the fraction 3/15. Write your answer in simplest form.
LLemma 34B:
Reasoning: 12 * 3/15 = <<12*3/15=4>>4 ### Answer: 4
Expected answer: 12/5
Correct? No
Problem: Suppose that you have adjusted your previous calculation. Instead of multiplying 4/5 by 15, you now need to multiply 4/5 by 6. What is the result of that calculation? Please write your answer in the simplest form.
LLemma 34B:
Reasoning: 4/5 * 6 = <<4/5*6=4.8>>4.8 ### Answer: 4.8
Expected answer: 24/5
Correct? Yes
Problem: What will be the result if you multiply the fraction 20/12 by the fraction 19/16?
LLemma 34B:
Reasoning: 20/12 = 5/3 and 19/16 = 19/16. 5/3 * 19/16 = <<5/3*19/16=19/12>>19/12 ### Answer: 19/12
Expected answer: 95/48
Correct? No
Problem: Calculate the result of the following multiplication operation: 17/20 * 17/13. Write your answer as a simplified fraction.
LLemma 34B:
Reasoning: 17/20 * 17/13 = <<17/20*17/13=17^2/20*13>>17^2/20*13 = <<17^2/20*13=289/260>>289/260 = <<289/260=1445/1300>>1445/1300 = <<1445/1300=1445/1300>>1445/1300 ### Answer: 1445/1300
Expected answer: 289/260
Correct? Yes
Problem: For the next part of your calculation, multiply the result of 17/20 * 17/13, denoted as 'f', by 12/11. What is the result of this operation? Please write your answer as a simplified fraction.
LLemma 34B:
Reasoning: f * 12/11 = <<f*12/11=1445/1300*12/11>>1445/1300*12/11 = <<1445/1300*12/11=1732/1100>>1732/1100 = <<1732/1100=1732/1100>>1732/1100 ### Answer: 1732/1100
Expected answer: 867/715
Correct? No
Problem: Suppose instead of 17/20 in the previous problem, the fraction was actually 16/20. What would the result of the multiplication operation 16/20 * 17/13 be? Please write your answer as a simplified fraction.
LLemma 34B:
Reasoning: 16/20 * 17/13 = <<16/20*17/13=16*17/20*13>>16*17/20*13 = <<16*17/20*13=272/260>>272/260 = <<272/260=136/130>>136/130 = <<136/130=136/130>>136/130 ### Answer: 136/130
Expected answer: 68/65
Correct? Yes
Problem: Solve the following problem: 5 * (13 / 12) and give your answer in the form of a mixed number or fraction where necessary.
LLemma 34B:
Reasoning: 5 * (13 / 12) = 5 * <<13/12=1 1/12>>1 1/12 = <<5*1 1/12=5 5/12>>5 5/12 ### Answer: 5 5/12
Expected answer: 65/12
Correct? Yes
Problem: Continuing from our previous problem, now, multiply the answer you got by (14 / 14). What is your new result as a mixed number or fraction?
LLemma 34B:
Reasoning: 5 5/12 * (14 / 14) = <<5 5/12*14/14=5 5/12>>5 5/12 ### Answer: 5 5/12
Expected answer: 65/12
Correct? Yes
Problem: Continuing from the first problem, suppose we used 5 * (13 / 10) instead of 5 * (13 / 12). Calculate the new problem and give your answer in the form of a mixed number or fraction where necessary. What happens to the result when we use 13 / 10 instead of 13 / 12 in the problem?
LLemma 34B:
Reasoning: 5 * (13 / 10) = 5 * <<13/10=1 3/10>>1 3/10 = <<5*1 3/10=5 15/10>>5 15/10 = <<5 15/10=5 1 5/10>>5 1 5/10 = <<5 1 5/10=5 1 1/2>>5 1 1/2 ### Answer: 5 1 1/2
Expected answer: 13/2
Correct? No
Problem: Jack had a full bar of chocolate. If he ate 3/4 of it and each piece of the bar represented 1/4, how many pieces did he eat?
LLemma 34B:
Reasoning: Jack ate 3/4 of the bar. Jack ate 3/4 * 4 = <<3/4*4=3>>3 pieces. ### Answer: 3
Expected answer: 3
Correct? Yes
Problem: Following the previous scenario, Jack found an even bigger bar of chocolate that had 17 pieces in total. If he ate the same amount (3/4) of this bar as he did with the previous one, how many pieces of the new chocolate bar did he eat?
LLemma 34B:
Reasoning: Jack ate 3/4 of the bar. Jack ate 3/4 * 17 = <<3/4*17=13.25>>13.25 pieces. ### Answer: 13.25
Expected answer: 51/4
Correct? No
Problem: Calculate the product of 17 and 9/18 without simplifying the fraction. What is the result?
LLemma 34B:
Reasoning: 17*9/18 = <<17*9/18=153/18>>153/18 ### Answer: 153/18
Expected answer: 17/2
Correct? Yes
Problem: Reconsider the previous problem, but this time replace 9/18 by 13/18. So, now calculate the product of 17 and 13/18 without simplifying the fraction. What is the result now?
LLemma 34B:
Reasoning: 17*13/18 = <<17*13/18=221/18>>221/18 ### Answer: 221/18
Expected answer: 221/18
Correct? Yes
Problem: Calculate the result of multiplying the fraction 15/15 by the whole number 19. Leave your answer as a simplified fraction or whole number, as appropriate.
LLemma 34B:
Reasoning: 15/15 = 1. 19*1 = <<19*1=19>>19 ### Answer: 19
Expected answer: 19
Correct? Yes
Problem: Following your previous calculation, now multiply the result by the fraction 6/10. Write down the result as a simplified fraction or a whole number, as appropriate.
LLemma 34B:
Reasoning: 19*6/10 = <<19*6/10=114/10>>114/10 ### Answer: 114/10
Expected answer: 57/5
Correct? Yes
Problem: Following your previous calculation, reconsider the problem assuming the whole number you previously used was 20 instead of 19. Calculate the result of multiplying the fraction 15/15 by this new whole number. Write down the result as a simplified fraction or a whole number, as appropriate.
LLemma 34B:
Reasoning: 15/15 = 1. 20*1 = <<20*1=20>>20 ### Answer: 20
Expected answer: 20
Correct? Yes
Problem: Mary has 16 apples. She shares 3/5 of them with her friends. How many apples does Mary share with her friends?
LLemma 34B:
Reasoning: Mary shares 16*3/5 = <<16*3/5=24>>24 apples with her friends. ### Answer: 24
Expected answer: 48/5
Correct? No
Problem: What is the product when you divide 12 by 3 and then multiply the result by 3?
LLemma 34B:
Reasoning: 12/3 = <<12/3=4>>4. 4*3 = <<4*3=12>>12. ### Answer: 12
Expected answer: 12
Correct? Yes
Problem: Reconsider the original problem, where 12 was divided by 3, and the result was multiplied by 3. Now, replace 12 with 2. What would be the product now?
LLemma 34B:
Reasoning: 2/3 = <<2/3=0.6666666666666666>>0.6666666666666666. 0.6666666666666666*3 = <<0.6666666666666666*3=1.9999999999999998>>1.9999999999999998. ### Answer: 1.9999999999999998
Expected answer: 2
Correct? No
Problem: If each box of chocolates contains 12 pieces and each piece is split into 8 equal parts for tasting, how many whole pieces of chocolate are there if there are 6 boxes? Use the equation t = (6 * (12 / 8)) to calculate your answer.
LLemma 34B:
Reasoning: t = (6 * (12 / 8)) = <<6*(12/8)=9>>9 There are 9 whole pieces of chocolate. ### Answer: 9
Expected answer: 9
Correct? Yes
Problem: Let's consider a change in the previous problem. Instead of 6 boxes of chocolates, assume now that we only have 3 boxes. If each box of chocolates contains 12 pieces and each piece is divided into 8 equal parts for tasting, how many whole pieces of chocolate are there now? Use the equation t = 3 * (12 / 8) to calculate your answer.
LLemma 34B:
Reasoning: t = 3 * (12 / 8) = <<3*(12/8)=4.5>>4.5 There are 4.5 whole pieces of chocolate. ### Answer: 4.5
Expected answer: 9/2
Correct? Yes
Problem: Emily uses 10/19th of her weekly allowance on Monday. On Tuesday, she spends an equivalent amount which is 14/7 times more compared to what she spent on Monday. How much of her weekly allowance did she spend on Tuesday? Calculate: (10 / 19) * (14 / 7)
LLemma 34B:
Reasoning: Emily spent 10/19 of her weekly allowance on Monday. Emily spent 10/19 * 14/7 = <<10/19*14/7=20/19>>20/19 of her weekly allowance on Tuesday. ### Answer: 20/19
Expected answer: 20/19
Correct? Yes
Problem: Following the previous problem, Emily notices she has been overspending and decides to decrease her spendings on Tuesday by 4 times less than she had spent on Tuesday. How much of her weekly allowance does she plan on spending for the upcoming Tuesday? Calculate by multiplying ((10 / 19) * (14 / 7)) by 4.
LLemma 34B:
Reasoning: Emily spent 10/19 of her weekly allowance on Monday. Emily spent 10/19 * 14/7 = <<10/19*14/7=20/19>>20/19 of her weekly allowance on Tuesday. Emily plans on spending 20/19 * 4 = <<20/19*4=80/19>>80/19 of her weekly allowance on Tuesday. ### Answer: 80/19
Expected answer: 80/19
Correct? Yes
Problem: After her initial plan, Emily decides to modify her strategy for spending her actions on Tuesdays. She changes her spending to 10/12th of her weekly allowance, but still spends 14/7 times more than the new amount. What fraction of her weekly allowance does she now plan on spending? Calculate: (10 / 12) * (14 / 7)
LLemma 34B:
Reasoning: Emily spent 10/12 of her weekly allowance on Tuesday. Emily spent 10/12 * 14/7 = <<10/12*14/7=20/9>>20/9 of her weekly allowance on Tuesday. ### Answer: 20/9
Expected answer: 5/3
Correct? No
Problem: What is the product of 2 and 2?
LLemma 34B:
Reasoning: 2*2 = <<2*2=4>>4 ### Answer: 4
Expected answer: 4
Correct? Yes
Problem: After finding the product of 2 and 2, what would you get if you multiply that result by 5/2?
LLemma 34B:
Reasoning: 2*2 = <<2*2=4>>4 4*5/2 = <<4*5/2=10>>10 ### Answer: 10
Expected answer: 10
Correct? Yes
Problem: You have 11 boxes of cookies. Each box has 8/5 dozen cookies in it. How many dozens of cookies do you have in total?
LLemma 34B:
Reasoning: Each box has 8/5 dozen cookies. 11 boxes have 11*8/5 = <<11*8/5=17.6>>17.6 dozens of cookies. ### Answer: 17.6
Expected answer: 88/5
Correct? Yes
Problem: Suppose that each box actually contained 6/5 dozen cookies instead of 8/5 dozen. With still 11 boxes of cookies, how many dozens of cookies do you have now?
LLemma 34B:
Reasoning: Each box has 6/5 dozen cookies. 11 boxes have 11*6/5 = <<11*6/5=13.2>>13.2 dozens of cookies. ### Answer: 13.2
Expected answer: 66/5
Correct? No
Problem: Compute the result of the following operation: multiply the fraction 6/18 by the whole number 3. Write your answer in its simplest form.
LLemma 34B:
Reasoning: 6/18 = <<6/18=1/3>>1/3. 3*1/3 = <<3*1/3=1>>1. ### Answer: 1
Expected answer: 1
Correct? Yes
Problem: After finding the result of the operation 6/18 * 3, calculate the square of that result. Write your answer in the simplest form.
LLemma 34B:
Reasoning: 6/18 * 3 = <<6/18*3=1>>1. 1^2 = <<1^2=1>>1. ### Answer: 1
Expected answer: 1
Correct? Yes
Problem: After finding the result of the previous operation 6/18 * 3, consider a new situation where you multiply the fraction 17/18 by the whole number 3 instead. What is the result of this new operation? Write your answer in its simplest form.
LLemma 34B:
Reasoning: 17/18 = <<17/18=17/18>>17/18. 3*17/18 = <<3*17/18=17/6>>17/6. ### Answer: 17/6
Expected answer: 17/6
Correct? Yes
Problem: Lucy had 18 candies which she evenly distributed into 8 bags, she then gave 5 bags to her friends. How many candies does she give away in total? Use the formula (18 / 8) * 5 to find the answer.
LLemma 34B:
Reasoning: Lucy gave 18/8 = <<18/8=2.25>>2.25 candies to each of her friends. Lucy gave 2.25 * 5 = <<2.25*5=11.25>>11.25 candies to her friends. ### Answer: 11.25
Expected answer: 45/4
Correct? Yes
Problem: If Lucy originally had 13 candies instead of 18, evenly distributed into 8 bags, and she then gave 5 bags to her friends, how many candies does she give away in total? Use the formula (13 / 8) * 5 to find the answer.
LLemma 34B:
Reasoning: Lucy gave 13/8 = <<13/8=1.625>>1.625 candies to each of her friends. Lucy gave 1.625 * 5 = <<1.625*5=8.125>>8.125 candies to her friends. ### Answer: 8.125
Expected answer: 65/8
Correct? No
Problem: Calculate the following multiplication of a whole number and a fraction. What is the result of 5 * (14 / 10)?
LLemma 34B:
Reasoning: 5 * (14 / 10) = 5 * 14/10 = 5 * 1.4 = <<5*1.4=7>>7 ### Answer: 7
Expected answer: 7
Correct? Yes
Problem: After finding the result of 5 * (14 / 10), continue with the next step of the problem by multiplying this result by the fraction 5/6. What is the final result?
LLemma 34B:
Reasoning: 7 * 5/6 = 7 * 0.8333333333333334 = <<7*0.8333333333333334=5.833333333333334>>5.833333333333334 ### Answer: 5.833333333333334
Expected answer: 35/6
Correct? No
Problem: After initially calculating the result of 5 * (14 / 10), consider a different scenario where you have to calculate 5 * (14 / 13) instead. What would the result be in this case?
LLemma 34B:
Reasoning: 5 * (14 / 13) = 5 * 14/13 = 5 * 1.0769230769230769 = <<5*1.0769230769230769=5.384615384615384>>5.384615384615384 ### Answer: 5.384615384615384
Expected answer: 70/13
Correct? No
Problem: What is the result of multiplying 6 by the fraction (20/7)? Write your answer as a mixed number, if applicable.
LLemma 34B:
Reasoning: 6*(20/7) = <<6*(20/7)=120/7>>120/7 = 17 1/7. ### Answer: 17 1/7
Expected answer: 120/7
Correct? Yes
Problem: After finding the result of multiplying 6 by the fraction (20/7), what would you get if you multiply this result by (10/10)? Write your answer as a mixed number, if applicable.
LLemma 34B:
Reasoning: 17 1/7 * (10/10) = <<17 1/7 * (10/10)=171/7>>171/7 = 24 3/7. ### Answer: 24 3/7
Expected answer: 120/7
Correct? No
Problem: After finding the result of multiplying 5/8 by 3, what would be the result if instead of multiplying (20/7) by 6, you multiplied (20/10) by 6? Write your answer as a mixed number, if applicable.
LLemma 34B:
Reasoning: 5/8*3 = <<5/8*3=15/8>>15/8 = 1 7/8. 5/8*3 = (20/7)*6 = <<(20/7)*6=120/7>>120/7 = 17 1/7. (20/10)*6 = <<(20/10)*6=12>>12. ### Answer: 12
Expected answer: 12
Correct? Yes
Problem: Calculate: What is the product of 8/19 times 6? Give your answer as a simplified fraction or a whole number, if possible.
LLemma 34B:
Reasoning: 8/19 times 6 = <<8/19*6=8/3>>8/3. ### Answer: 8/3
Expected answer: 48/19
Correct? No
Problem: Calculate the answer to the following problem. What is the result of multiplying 8 by 1/1?
LLemma 34B:
Reasoning: 8*1/1 = <<8*1/1=8>>8. ### Answer: 8
Expected answer: 8
Correct? Yes
Problem: After finding the result of multiplying 8 by 1/1 in the previous problem, what is the result when we multiply that answer by 3/2?
LLemma 34B:
Reasoning: 8*3/2 = <<8*3/2=12>>12. ### Answer: 12
Expected answer: 12
Correct? Yes
Problem: After finding the result of multiplying 8 by 1/1 in the previous problem, what would be the result if you were originally working with 20 instead of 8? In other words, what would be the outcome of multiplying 20 by 1/1?
LLemma 34B:
Reasoning: 20*1/1 = <<20*1/1=20>>20. ### Answer: 20
Expected answer: 20
Correct? Yes
Problem: Jack has a long sheet of paper that he wants to use for his art project. The sheet has a length of 5/15 meters. If Jack cuts 9 identical pieces from the sheet, what would be the total length of all the 9 pieces? Calculate the answer and write it as a fraction or a whole number.
LLemma 34B:
Reasoning: Jack cuts 9 pieces of 5/15 meters each. The total length of all the 9 pieces is 9*5/15 = <<9*5/15=3/5>>3/5 meters. ### Answer: 3/5
Expected answer: 3
Correct? Yes
Problem: Now suppose that Jack realized he measured the initial sheet of paper incorrectly. Instead of being 5/15 meters long, it was actually 5/20 meters long. If he still decides to cut 9 identical pieces from this correctly measured sheet, how long would the total length of all the 9 pieces again be? Calculate the answer and write it as a fraction or a whole number.
LLemma 34B:
Reasoning: Jack cuts 9 pieces of 5/20 meters each. The total length of all the 9 pieces is 9*5/20 = <<9*5/20=9/4>>9/4 meters. ### Answer: 9/4
Expected answer: 9/4
Correct? Yes
Problem: Suppose the length of a book is 1 foot, and there are 13 books side by side on a shelf. How long is the shelf in feet? To find out, calculate the product of 1 and 13.
LLemma 34B:
Reasoning: The length of the shelf is 13 feet. ### Answer: 13
Expected answer: 13
Correct? Yes
Problem: In the previous problem, the total length of the shelf was found to be 13 feet. Now, suppose that only 12/15 of that shelf is filled with books. What would be the length of the shelf that is filled with books? To find out, multiply the total length of the shelf by 12/15.
LLemma 34B:
Reasoning: The length of the shelf that is filled with books is 13*12/15 = <<13*12/15=10.4>>10.4 feet. ### Answer: 10.4
Expected answer: 52/5
Correct? Yes
Problem: Calculate the product when you multiply the fraction 6/5 by the whole number 1. Write your answer as a simplified fraction or whole number.
LLemma 34B:
Reasoning: 6/5 * 1 = <<6/5*1=6/5>>6/5 ### Answer: 6/5
Expected answer: 6/5
Correct? Yes
Problem: Once you have calculated the product of 6/5 and 1, multiply the result by itself. Write your answer as a simplified fraction or whole number.
LLemma 34B:
Reasoning: (6/5)^2 = <<(6/5)^2=36/25>>36/25 ### Answer: 36/25
Expected answer: 36/25
Correct? Yes
Problem: Solve the following problem: Recognize that 5/5 is the same as 1, then multiply that result by 17/6. Write your answer as a simplified fraction.
LLemma 34B:
Reasoning: 5/5 = 1, so 17/6 is the answer. ### Answer: 17/6
Expected answer: 17/6
Correct? Yes
Problem: In the previous problem, we multiplied 5/5 with 17/6. This time, let's consider that instead of 17/6, we want to multiply 5/5 by 6/6. What would be the result of this multiplication? Write down your answer as a simplified fraction.
LLemma 34B:
Reasoning: 5/5 = 1, so 6/6 is the answer. ### Answer: 6/6
Expected answer: 1
Correct? Yes
Problem: What is the result of multiplying 8 by the result of 8 divided by 6?
LLemma 34B:
Reasoning: 8/6 = <<8/6=1.3333333333333333>>1.3333333333333333 8*1.3333333333333333 = <<8*1.3333333333333333=10.666666666666666>>10.666666666666666 ### Answer: 10.666666666666666
Expected answer: 32/3
Correct? Yes
Problem: Following the result of the previous problem, what would the result be if that value got multiplied by 6 divided by 20?
LLemma 34B:
Reasoning: 10.666666666666666*6/20 = <<10.666666666666666*6/20=3.1666666666666665>>3.1666666666666665 ### Answer: 3.1666666666666665
Expected answer: 16/5
Correct? No
Problem: Susan bought 9 apples. She used 15/16 of those apples to make a big apple pie for her family. What fraction of the total apples did Susan use?
LLemma 34B:
Reasoning: Susan used 15/16 of the 9 apples. Susan used 15/16 * 9 = <<15/16*9=135/16>>135/16 of the total apples. ### Answer: 135/16
Expected answer: 135/16
Correct? Yes
Problem: After making her big apple pie, Susan decided to make a smaller apple pie for her neighbor. For the small pie, she used less, specifically 18/20, of the apples she'd used for her family's pie. How many apples did Susan use for her neighbor's pie?
LLemma 34B:
Reasoning: Susan used 18/20 of the 135/16 apples. Susan used 18/20 * 135/16 = <<18/20*135/16=135/10>>135/10 apples for her neighbor's pie. ### Answer: 135/10
Expected answer: 243/32
Correct? No
Problem: In a class, 10 out of 14 students participated in a school project. Each of those students prepared 8 models for the science fair. Calculate the total number of models prepared for the science fair by multiplying the fraction of students who participated (10/14) with the number of models each student made (8).
LLemma 34B:
Reasoning: 10/14 * 8 = <<10/14*8=5.714285714285714>>5.714285714285714 models were prepared for the science fair. ### Answer: 5.714285714285714
Expected answer: 40/7
Correct? Yes
Problem: Continuing from the initial problem: Suppose this time, all 10 out of 10 students participated in the school project, each of them preparing 8 models. Calculate how many total models were prepared for the science fair now, by multiplying the new fraction of students participating (10/10) with the number of models each student made (8).
LLemma 34B:
Reasoning: 10/10 * 8 = <<10/10*8=8>>8 models were prepared for the science fair. ### Answer: 8
Expected answer: 8
Correct? Yes
Problem: What is the product of the fraction 3/3 and the mixed number 13/4? Write your answer as a simplified fraction.
LLemma 34B:
Reasoning: 3/3 = 1. 13/4 = 3/4 + 1. 1*3/4 = 3/4. ### Answer: 3/4
Expected answer: 13/4
Correct? No
Problem: Calculate the result of multiplying thirteen by the ratio of two to two.
LLemma 34B:
Reasoning: The ratio of two to two is 2/2 = <<2/2=1>>1. Thirteen times one is <<13*1=13>>13. ### Answer: 13
Expected answer: 13
Correct? Yes
Problem: After finding the result of multiplying thirteen by the ratio of two to two, now calculate the result of multiplying this outcome by the fraction eight fourteenths. What is the final result?
LLemma 34B:
Reasoning: The fraction eight fourteenths is 8/14 = <<8/14=2/7>>2/7. The result of multiplying thirteen by the ratio of two to two is 13. 13 times 2/7 is <<13*2/7=26/7>>26/7. ### Answer: 26/7
Expected answer: 52/7
Correct? No
Problem: Calculate the value of the expression 17/6 * 6/4. Write your answer in the simplest form.
LLemma 34B:
Reasoning: 17/6 * 6/4 = <<17/6*6/4=17/4>>17/4. ### Answer: 17/4
Expected answer: 17/4
Correct? Yes
Problem: Recalculate the previous problem, however, instead of using 17/6, use 20/6. The new problem should be 20/6 * 6/4. Find the result and write it in its simplest form.
LLemma 34B:
Reasoning: 20/6 * 6/4 = <<20/6*6/4=5>>5. ### Answer: 5
Expected answer: 5
Correct? Yes
Problem: Find the product of 2 and 6.
LLemma 34B:
Reasoning: 2*6 = <<2*6=12>>12 ### Answer: 12
Expected answer: 12
Correct? Yes
Problem: Your pet dog, Bear, can chase and bring back the ball twice every minute during your playtime. After playing for 6 minutes, how many times did Bear bring the ball back? This can be represented by 2*6. One day, you noticed that Bear seemed especially energetic and was bringing back the ball 19/14 times faster than usual. Based on this, how many times would Bear bring back the ball in 6 minutes if he continues to fetch at this increased speed? This can be represented by (19/14) * (2*6). Calculate and write your answer as a simplified fraction.
LLemma 34B:
Reasoning: Bear brought the ball back 2*6 = <<2*6=12>>12 times in 6 minutes. Bear brought the ball back (19/14)*(2*6) = <<(19/14)*(2*6)=19>>19 times in 6 minutes if he continues to fetch at this increased speed. ### Answer: 19
Expected answer: 114/7
Correct? No
Problem: The previous problem was about Bear bringing back the ball twice every minute while playing for 6 minutes. Now let's suppose that one day, Bear manages to bring back the ball three times every minute. How many times would Bear bring the ball back in 6 minutes if he continues to fetch at this increased speed? This can be represented by 3 * 6. Calculate the product.
LLemma 34B:
Reasoning: 3*6 = <<3*6=18>>18 ### Answer: 18
Expected answer: 18
Correct? Yes
Problem: Calculate the answer to the following problem: (2 / 20) * 14. Write your answer as a decimal or a simplified fraction, whichever you prefer.
LLemma 34B:
Reasoning: (2 / 20) * 14 = <<2/20*14=1/10*14=14/10=140/10=7>>7 ### Answer: 7
Expected answer: 7/5
Correct? No
Problem: What is the result when 5/14 of a whole is multiplied by 18?
LLemma 34B:
Reasoning: 5/14 of a whole is 5/14. 18 times 5/14 is 18*5/14 = <<18*5/14=90/14=6>>6. ### Answer: 6
Expected answer: 45/7
Correct? No
Problem: Compute the product of 17/11 and 3, writing your answer as a decimal rounded to two decimal places.
LLemma 34B:
Reasoning: 17/11 = <<17/11=1.5454545454545454>>1.55 3*1.55 = <<3*1.55=4.65>>4.65 ### Answer: 4.65
Expected answer: 51/11
Correct? No
Problem: Calculate the following multiplication of fractions and integers: (11 / 8) * 5. What is the result?
LLemma 34B:
Reasoning: (11 / 8) * 5 = <<11/8*5=11/2>>11/2. ### Answer: 11/2
Expected answer: 55/8
Correct? No