seegongsik
Grade 5 (age 10-11)

grade 5 Fraction Multiplication & Division

To multiply fractions, multiply the numerators together and the denominators together; it is the same as finding a fraction of an amount. If you eat 3/4 of the 2/3 of a cake that is left, you have eaten 2/3 × 3/4 = 6/12 = 1/2 of the whole cake. To divide by a fraction, multiply by its reciprocal, the fraction with its numerator and denominator swapped. The first picture shows multiplication as the area of a rectangle; the second shows division as cutting a ribbon into pieces.

Multiplying fractions: the area model

🟩 The overlap is the product
①Cut the rectangle into rows by the first denominator and into columns by the second, and you get 3 × 4 = 12 cells
②Shade the first fraction of the rows blue from the top and the second fraction of the columns orange from the left: the cells where both colors overlap turn green, and there are 2 × 3 = 6 of them
③So 2/3 × 3/4 = 6/12, which simplifies to 1/2. Multiplying the numerators counts the green cells, and multiplying the denominators counts all the cells
④When both fractions are less than 1, the green part is smaller than the blue part and smaller than the orange part
2
3
3
4

Dividing fractions: how many fit?

📏 Division counts the pieces
①Cut a 2/3 m ribbon into 3/4 m pieces and you cannot fill even one piece. The green part is 8/9 of a piece, so 2/3 ÷ 3/4 = 8/9
②Change the second fraction to 1/4 and two orange pieces fill up, with a green part 2/3 of a piece long left over, so the quotient is 2 2/3, which is 8/3
③Dividing means counting how many piece lengths fit into the whole length, and that count equals multiplying by the flipped fraction
④Dividing by a number less than 1 gives a quotient bigger than the number you started with
↔️ The same two fractions
The two fractions you chose above are used again below: the first is the whole length and the second is the length of one piece.

Summary

Fraction Multiplication
ab × cd = a×cb×d
Multiply the numerators together and the denominators together
Fraction Division
ab ÷ cd = ab × dc
Swap the numerator and denominator of the divisor (its reciprocal) and multiply
🎯 At a glance
①Multiplication: numerator times numerator, denominator times denominator
②In the area model, the product is green cells ÷ all cells
③Division: flip the divisor and multiply
④Multiplying by a number less than 1 makes the result smaller, and dividing by one makes it larger
⑤After calculating, simplify to lowest terms

Examples and Unit Test

Example 1
Work out 2/3 × 3/4 and simplify.
1
Multiply the numerators and the denominators: (2 × 3)/(3 × 4) = 6/12.
2
Divide top and bottom by their GCD, 6, to get 1/2.
▸ 12
This is the starting setting of the first picture: 6 of the 12 cells, exactly half, are green.
Example 2
Work out 2/3 ÷ 1/4.
1
Flip the divisor 1/4 to 4/1 and multiply: 2/3 × 4/1.
2
(2 × 4)/(3 × 1) = 8/3.
▸ 83
Set the second fraction to 1/4 in the second picture and you see two orange pieces plus a green part 2/3 of a piece long, so 8/3 = 2 2/3.
Unit Test
What is 1/2 × 4/5?
①58
②25
③410
④47
▸ ② 25
1
(1 × 4)/(2 × 5) = 4/10.
2
Simplify 4/10 by the GCD, 2, to get 2/5.
← Previous
Reduction & Common Denominator
Next →
Decimal Multiplication & Division

Last updated