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Math · Fractions
Multiply Fractions: Whole Numbers, Fractions of Quantities, Estimate, Check
Move from fraction addition into multiplication: fractions of whole-number quantities, fraction × fraction, mixed numbers, simplification, reverse problems and reasonableness checks.
The idea
Multiplying a fraction by a whole number can mean repeated groups or a fraction of a quantity. Three groups of 1/4 make 3/4, while 3/5 of 20 means split 20 into 5 equal parts and take 3 of them.
For fraction × fraction, the product represents part of a part. Multiply numerators and denominators, then simplify. You may simplify common factors before multiplying when that keeps the arithmetic smaller.
Estimate first. A proper fraction is less than 1, so multiplying a positive number by a proper fraction should make it smaller. That size check catches many numerator/denominator mistakes before they spread.
Worked examples
Example 1 · Fraction × whole number
Find 3 × 1/4.
- Three groups of one fourth make 1/4 + 1/4 + 1/4.
- Add the fractional parts: 3/4.
- The product is still less than 1, which fits the estimate.
3 × 1/4 = 3/4.
Example 2 · Fraction of a quantity
Find 3/5 of 20.
- Divide 20 into 5 equal parts: 20 ÷ 5 = 4.
- Take 3 of those parts: 3 × 4 = 12.
- Equivalently, 3/5 × 20 = 12.
3/5 of 20 = 12.
Example 3 · Fraction × fraction
Find 2/3 × 3/4.
- Multiply numerators: 2 × 3 = 6.
- Multiply denominators: 3 × 4 = 12.
- Simplify 6/12 to 1/2.
2/3 × 3/4 = 1/2.
Example 4 · Simplify before multiplying
Find 7/12 × 8/21.
- Cancel common factors before multiplying: 7 with 21 leaves 1 and 3; 8 with 12 leaves 2 and 3.
- Now multiply 1 × 2 over 3 × 3.
- The smaller arithmetic gives 2/9.
7/12 × 8/21 = 2/9.
Example 5 · Mixed number
Find 1 3/4 × 2/7.
- Convert 1 3/4 to 7/4.
- Multiply 7/4 × 2/7.
- Cancel 7 with 7 and simplify 2/4 to 1/2.
1 3/4 × 2/7 = 1/2.
Example 6 · Reverse problem
3/5 of x is 18. Find x.
- Write (3/5)x = 18.
- Undo multiplication by 3/5 by dividing by 3/5, or multiply by 5/3.
- x = 18 × 5/3 = 30.
- Check: 3/5 of 30 is 18.
x = 30.
Common mistakes
These are tempting because they use familiar operations. Check what the question is actually measuring.
Why this fails: Both the number of parts and the size of each part change. Multiply both numerators and denominators, then simplify: 8/15.
Why this fails: The word 'of' signals multiplication here. Divide 24 by 4, then take 3 parts: 18.
Why this fails: Multiplying a positive number by a proper fraction makes it smaller because you are taking only part of it.
Why this fails: Cross-cancellation works only on common factors in a multiplication expression, never across addition or subtraction.
Why this fails: For 1 3/4, compute 1 × 4 + 3 = 7, so the improper fraction is 7/4.
Foundation
Foundation practice
Connect repeated groups, fractions of quantities and simple fraction products.
1. 3 × 1/4
2. 5 × 2/3
3. Find 3/5 of 20.
4. Find 7/8 of 32.
5. 2/3 × 9
6. 4/7 × 14
7. 1/2 × 3/5
8. 2/3 × 3/4
9. 5/6 × 3/10
10. 7/8 × 4/7
Core
Core practice
Multiply, simplify and work with mixed numbers.
11. 3/4 × 2/5
12. 5/9 × 6/7
13. 7/12 × 8/21
14. 11/15 × 5/22
15. Find 4/5 of 35.
16. Find 3/8 of 56.
17. 2/3 × 1 1/2
18. 1 3/4 × 2/7
19. 2 1/3 × 3/5
20. 1 1/4 × 2 2/5
Think
Think practice
Estimate product size, analyse errors, compare expressions and solve reverse problems.
21. Estimate 5/6 × 7/8, then find the exact product.
22. Estimate 1 7/8 × 2 2/3, then find the exact product.
23. A student says 2/3 × 4/5 = 8/8. Explain the error.
24. A student says 3/4 of 24 means 3 ÷ (4 × 24). What should happen?
25. 3/5 of x is 18. Find x.
26. 2/3 of x is 14. Find x.
27. x × 5/8 = 15/16. Find x.
28. Which is greater: 3/4 × 2/3 or 5/6 × 3/5?
29. Which is greater: 2/5 of 30 or 3/8 of 32?
30. Without multiplying first, explain why 4/5 × 3/4 must be less than 4/5.
Challenge
Challenge practice
Use equations, mixed numbers and multi-step contexts as a bridge toward prealgebra.
31. 7/9 × 27
32. 5/12 × 36
33. 2 1/2 × 3/5
34. 1 2/3 × 2 1/4
35. (2/7)x = 10. Find x.
36. (3/4)x = 2 1/4. Find x.
37. A recipe needs 2/3 of a 3/4-cup measure. How much is that?
38. A project uses 5/6 of a 2 2/5 m ribbon. How much ribbon is used?
39. A box has 40 counters. 3/5 are blue. One quarter of the blue counters are dark blue. How many are dark blue?
40. Why does multiplying a/b by c/d give (a×c)/(b×d)?
Answer key
- 1. 3/4
Three groups of 1/4 make 3/4. - 2. 3 1/3
10/3 = 3 1/3. - 3. 12
20 ÷ 5 = 4, and 3 × 4 = 12. - 4. 28
32 ÷ 8 = 4, and 7 × 4 = 28. - 5. 6
9 ÷ 3 = 3, then 2 × 3 = 6. - 6. 8
14 ÷ 7 = 2, then 4 × 2 = 8. - 7. 3/10
Multiply 1 × 3 over 2 × 5. - 8. 1/2
6/12 simplifies to 1/2. - 9. 1/4
15/60 simplifies to 1/4. - 10. 1/2
28/56 simplifies to 1/2. - 11. 3/10
6/20 simplifies to 3/10. - 12. 10/21
30/63 simplifies by 3 to 10/21. - 13. 2/9
56/252 simplifies to 2/9; cancelling first makes the arithmetic easier. - 14. 1/6
55/330 simplifies to 1/6. - 15. 28
35 ÷ 5 = 7; 4 × 7 = 28. - 16. 21
56 ÷ 8 = 7; 3 × 7 = 21. - 17. 1
1 1/2 = 3/2, so 2/3 × 3/2 = 1. - 18. 1/2
7/4 × 2/7 simplifies to 2/4 = 1/2. - 19. 1 2/5
7/3 × 3/5 = 7/5 = 1 2/5. - 20. 3
5/4 × 12/5 = 60/20 = 3. - 21. Estimate: about 3/4; exact: 35/48.
Both factors are below 1, so the product must be below either factor. 35/48 is close to 3/4. - 22. Estimate: about 5; exact: 5.
15/8 × 8/3 = 5, matching the estimate near 2 × 2.5. - 23. The correct product is 8/15.
Multiply denominators too: 3 × 5 = 15. - 24. 3/4 of 24 = 18.
Divide 24 into fourths first: 24 ÷ 4 = 6, then take 3 parts. - 25. x = 30.
18 ÷ (3/5) = 18 × 5/3 = 30. - 26. x = 21.
14 × 3/2 = 21. - 27. x = 1 1/2.
15/16 ÷ 5/8 = 15/16 × 8/5 = 3/2. - 28. They are equal.
Each product simplifies to 1/2. - 29. They are equal at 12.
2/5 of 30 = 12 and 3/8 of 32 = 12. - 30. Because 3/4 is less than 1, taking 3/4 of 4/5 keeps only part of 4/5.
The exact product is 3/5, confirming the size reasoning. - 31. 21
27 ÷ 9 = 3, then 7 × 3 = 21. - 32. 15
36 ÷ 12 = 3, then 5 × 3 = 15. - 33. 1 1/2
5/2 × 3/5 = 3/2. - 34. 3 3/4
5/3 × 9/4 = 45/12 = 15/4. - 35. x = 35.
Multiply 10 by 7/2. - 36. x = 3.
2 1/4 = 9/4; dividing by 3/4 gives 3. - 37. 1/2 cup.
2/3 × 3/4 = 6/12 = 1/2. - 38. 2 m.
5/6 × 12/5 = 2. - 39. 6
3/5 of 40 is 24; 1/4 of 24 is 6. - 40. Because taking c/d of a/b divides the amount into d equal parts and keeps c of them, so both the count of selected parts and the size of each part are represented in the numerator and denominator products.
The rule records a 'part of a part' relationship rather than being an isolated trick.