Multiplying a Fraction by a Whole Number - Fourth Grade Mathematics
When you cook four batches of brownies and each batch calls for 2/3 of a cup of cocoa powder, how much cocoa powder do you need in total? In primary school, you learned that multiplication represents equal groups: 4 groups of 2 cups is 4 times 2. In fourth grade, that exact same principle applies directly to fractions! Multiplying a whole number by a fraction is simply repeated addition of equal fractional amounts. In this chapter, you will connect visual area models and number-line jumps to multiplication, master the algebraic rule for multiplying a whole number by a fraction, and discover how to express the product as both an improper fraction and a mixed number.
The Foundation: Repeated Addition of Equal Fractional Groups
Just as 4 x 3 means 3 + 3 + 3 + 3, multiplying a whole number by a fraction means adding that fraction repeatedly.
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| VISUALIZING 4 x 2/3 AS REPEATED ADDITION |
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| Expression: 4 x (2/3) |
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| Repeated Addition: 2/3 + 2/3 + 2/3 + 2/3 |
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| Combine numerators over common denominator: |
| (2 + 2 + 2 + 2) / 3 = 8/3 |
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| Visual Number Line Model: |
| 0 1/3 2/3 1 4/3 5/3 2 7/3 8/3 3 |
| +----------+----------+----+-----+----------+----+-----+----------+----+ |
| [-- +2/3 --] |
| [-- +2/3 --] |
| [-- +2/3 --] |
| [-- +2/3 --] |
| LAND AT 8/3! |
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| Convert 8/3 to a mixed number: 8 / 3 = 2 with remainder 2 = 2 and 2/3! |
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The Associative Property and Unit Fractions
Notice the profound connection to unit fractions! 2/3 is equal to 2 x (1/3). Therefore, 4 x (2/3) can be rewritten as: 4 x (2 x 1/3) = (4 x 2) x (1/3) = 8 x (1/3) = 8/3. You are simply multiplying the whole number by the numerator, and the result counts how many unit fractions (1/3) you possess!
The Universal Multiplication Rule
To multiply a whole number by a fraction, multiply the whole number by the numerator, while keeping the denominator unchanged.
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| WHOLE NUMBER BY FRACTION FORMULA |
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| Formula: w x (a / b) = (w x a) / b |
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| Example: 5 x 3/4 |
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| Step 1: Multiply whole number by numerator: |
| 5 x 3 = 15 |
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| Step 2: Place over the denominator: |
| 15/4 |
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| Step 3: Convert to a mixed number: |
| 15 / 4 = 3 with remainder 3 = 3 and 3/4 |
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Why the Denominator Is Not Multiplied
Why don't we multiply 5 by 4 in the denominator? Because a whole number w can be written as a fraction with a denominator of 1: w = w/1! 5 = 5/1. When multiplying fractions: (5/1) x (3/4) = (5 x 3) / (1 x 4) = 15/4. Multiplying the denominator by 5 would give 15/20, which is equal to 3/4—meaning 5 batches of 3/4 would equal only 3/4! That makes no sense. The denominator must stay 4.
Fraction of a Set: Multiplication as "Of"
In word problems, the word "of" almost always indicates multiplication: "Find 3/4 of 20."
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| FINDING 3/4 OF 20 |
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| Method 1 (Multiply then Divide): |
| (3 x 20) / 4 = 60 / 4 = 15 |
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| Method 2 (Divide then Multiply): |
| Find 1/4 of 20: 20 / 4 = 5 |
| Now take 3 of those fourths: 3 x 5 = 15 |
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| Both methods show that 3/4 of 20 is exactly 15! |
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Chapter Practice Exercises
Exercise 1: Compute 6 x 3/5. Write the answer as both an improper fraction and a mixed number.
Exercise 2: Write 5 x 2/7 as a repeated addition sentence, and find the product.
Exercise 3: Compute 8 x 3/4. Show both the improper fraction and the simplified whole number result.
Exercise 4: A pet shelter feeds each puppy 3/8 pound of food each day. If there are 6 puppies, how many pounds of food are needed each day?
Exercise 5: A student calculates 3 x 2/5 and writes 6/15. Explain the student's misconception and calculate the correct product.
Solutions and Step-by-Step Explanations
Solution 1: We multiply the whole number by the numerator: 6 x 3 = 18. Placing over the denominator gives 18/5. To convert to a mixed number, divide 18 by 5: 18 / 5 = 3 with a remainder of 3. The product is 18/5 or 3 and 3/5.
Solution 2: Expressed as repeated addition: 5 x 2/7 = 2/7 + 2/7 + 2/7 + 2/7 + 2/7. Adding the numerators: (2 + 2 + 2 + 2 + 2) / 7 = 10/7, which equals 1 and 3/7.
Solution 3: Multiplying the whole number by the numerator: 8 x 3 = 24. Placing over the denominator gives 24/4. Dividing 24 by 4 gives exactly 6 wholes (6).
Solution 4: To find the total food needed, multiply 6 puppies by 3/8 pound: 6 x 3/8 = (6 x 3) / 8 = 18/8 pounds. Converting to a mixed number: 18 / 8 = 2 with remainder 2, which is 2 and 2/8 pounds = 2 and 1/4 pounds. The shelter needs 2 and 1/4 pounds of food each day.
Solution 5: The student mistakenly multiplied both the numerator and the denominator by 3: (3 x 2) / (3 x 5) = 6/15. Multiplying top and bottom by 3 creates an equivalent fraction, leaving the value at 2/5! Three groups of 2/5 must be larger than 2/5. The whole number 3 only multiplies the numerator: (3 x 2) / 5 = 6/5 = 1 and 1/5.