Multiplying Fractions

Study Sheet

Multiplying Fractions

Straight across, and why

Why Straight Across Works

Concept
The Big Idea
1312
One third across, one half down: the doubly shaded overlap is 1 of 6 equal pieces.

Multiplying fractions means taking a part of a part. Cutting a rectangle both ways shows why you multiply tops and bottoms separately.

Multiplying by a fraction means taking a part of something: 12×13\tfrac12\times\tfrac13 is half of a third. Multiply the tops and multiply the bottoms — the picture is a rectangle cut both ways, and the doubly-shaded piece is the answer. Multiplying by a fraction less than 11 makes things smaller, which surprises people who expect multiplication to grow.

Example
Finding 23×34\dfrac{2}{3} \times \dfrac{3}{4}
2334
3 × 4 = 12 pieces in all; 2 × 3 = 6 of them are doubly shaded: 6/12 = 1/2.

Straight across: 2×33×4=612=12\dfrac{2 \times 3}{3 \times 4} = \dfrac{6}{12} = \dfrac{1}{2}.

Example
A fraction of a whole number: 23\tfrac23 of 1212
44412
Twelve in three equal parts; two parts shaded.

Split 1212 into 33 equal parts: each part is 12÷3=412 \div 3 = 4. Two parts are 2×4=82 \times 4 = 8. As a multiplication: 23×12=2×123=243=8\tfrac23 \times 12 = \tfrac{2 \times 12}{3} = \tfrac{24}{3} = 8.

Side note
Where it goes wrong

Finding a common denominator before multiplying. That is the rule for adding; for multiplying you go straight across.

Tip
It gets smaller

Multiplying by a fraction less than 11 makes the result smaller — surprising at first, but taking part of something gives less than you started with.

Try it
Try it: straight across

Find 34×25\dfrac34 \times \dfrac25 and simplify.

Show the answer

Straight across: 3×24×5=620\dfrac{3 \times 2}{4 \times 5} = \dfrac{6}{20}. Divide top and bottom by 22: 310\mathbf{\dfrac{3}{10}}.

In Plain Terms & More Examples

Concept
In plain terms

Multiply the top numbers together and the bottom numbers together, then simplify. Taking a part of a part gives a smaller result.

Example
Finding 35×56\dfrac{3}{5} \times \dfrac{5}{6}
3556
15 of the 30 small pieces are doubly shaded: 15/30 = 1/2.

Straight across: 3×55×6=1530=12\dfrac{3 \times 5}{5 \times 6} = \dfrac{15}{30} = \dfrac{1}{2}.

Example
Finding 12×23\dfrac{1}{2} \times \dfrac{2}{3}

Straight across: 1×22×3=26=13\dfrac{1 \times 2}{2 \times 3} = \dfrac{2}{6} = \dfrac{1}{3}.

Side note
Where it goes wrong

Multiplying across the diagonal. For 35×56\tfrac35 \times \tfrac56, multiplying each top by the other bottom gives 3×65×5=1825\tfrac{3 \times 6}{5 \times 5} = \tfrac{18}{25}, which is more than 35\tfrac35. But 56\tfrac56 of 35\tfrac35 is a part of 35\tfrac35, so it must be less than 35\tfrac35. Tops go with tops, bottoms with bottoms.

Try it
Try it: simplify at the end

Find 25×58\dfrac25 \times \dfrac58.

Show the answer

2×55×8=1040=14\dfrac{2 \times 5}{5 \times 8} = \dfrac{10}{40} = \mathbf{\dfrac14}.

Mixed Numbers and Whole Numbers

Concept
Convert first

Turn mixed numbers into improper fractions before multiplying: 112=321\tfrac12 = \tfrac32. A whole number is a fraction over 11.

A whole number is a fraction with bottom 11: 4=414 = \tfrac41. A mixed number has to be made improper first because the rule "tops times tops, bottoms times bottoms" works only on a single fraction, and 2132\tfrac13 is really a sum, 2+132 + \tfrac13.

Example
Finding 213×352\tfrac13 \times \tfrac35

73×35=2115=75=125\tfrac73\times\tfrac35 = \tfrac{21}{15} = \tfrac75 = 1\tfrac25.

Example
Whole number times a fraction: 4×384 \times \tfrac38
012132238383838
Four jumps of 3/8 end at 12/8 = 1 1/2.

4×384 \times \tfrac38 is 44 jumps of 38\tfrac38: 38+38+38+38=128\tfrac38 + \tfrac38 + \tfrac38 + \tfrac38 = \tfrac{12}{8}. Straight across, with 4=414 = \tfrac41: 41×38=128=148=112\tfrac41 \times \tfrac38 = \tfrac{12}{8} = 1\tfrac48 = 1\tfrac12.

Side note
Where it goes wrong
221212111412
The shortcut keeps only 2 and 1/4 and forgets the two cross pieces, 1 and 1/2.

Multiplying the whole parts and the fraction parts separately. 212×1122\tfrac12 \times 1\tfrac12 is not 2×1+12×12=2142 \times 1 + \tfrac12 \times \tfrac12 = 2\tfrac14. The area model has four pieces, not two: 2×1=22 \times 1 = 2, 2×12=12 \times \tfrac12 = 1, 12×1=12\tfrac12 \times 1 = \tfrac12 and 12×12=14\tfrac12 \times \tfrac12 = \tfrac14, which total 3343\tfrac34. Converting first agrees: 52×32=154=334\tfrac52 \times \tfrac32 = \tfrac{15}{4} = 3\tfrac34.

Try it
Try it: cancel before you multiply

Compute 49×38\tfrac{4}{9}\times\tfrac{3}{8} by cancelling common factors first.

Show the answer

Cancel 44 with 88 and 33 with 99: 13×12=16\tfrac{1}{3}\times\tfrac{1}{2} = \mathbf{\tfrac16}.

Word Problems

Concept
"Of" means times

34\tfrac34 of something is 34×\tfrac34 \times that thing. When a story takes a fraction of a fraction, multiply the two fractions.

Taking 34\tfrac34 of an amount means cutting it into 44 equal parts and keeping 33, which is exactly what multiplying by 34\tfrac34 does. So "of" in a fraction story is a signal to multiply, the way "each" is in a whole-number story.

Example
Part of a part
girlssoccer
Girls are 2 of 3 parts; 3/4 of each girls’ part plays soccer: 6 of 12 twelfths, half the class.

Two thirds of a class are girls and 34\tfrac34 of the girls play soccer. The fraction of the class that are soccer-playing girls: 34×23=612=12\tfrac34\times\tfrac23 = \tfrac{6}{12} = \tfrac12.

Example
More or less?

A path is 55 km long and Kai has walked 34\tfrac34 of it. Before calculating: 34\tfrac34 is less than 11, so Kai has walked less than 55 km. Calculating: 34×5=154=334\tfrac34 \times 5 = \tfrac{15}{4} = 3\tfrac34 km.

Side note
Where it goes wrong

Adding when the story says "of". "34\tfrac34 of the girls" is not 34+23\tfrac34 + \tfrac23; that would be 1712\tfrac{17}{12}, more than the whole class. A part of a group can never be bigger than the group.

Try it
Try it: the recipe

A recipe needs 34\tfrac34 cup of sugar. You make 2122\tfrac12 batches. How much sugar?

Show the answer

52×34=158=178\tfrac52\times\tfrac34 = \tfrac{15}{8} = \mathbf{1\tfrac78} cups.

Multiplying as Scaling

Concept
Bigger or smaller?

Multiplying by a number greater than 11 gives more than you started with; multiplying by a number less than 11 gives less; multiplying by exactly 11 changes nothing.

Think of multiplying as resizing. 8×548 \times \tfrac54 is 88 plus another quarter of 88, so it is more than 88. 8×348 \times \tfrac34 is 88 minus a quarter of 88, so it is less. You can tell which way before doing any arithmetic, which makes a quick check on every answer.

Example
Scaling 88
= 68 ×34= 88 ×1= 108 ×54
The dashed line marks 8: one bar falls short, one reaches it, one passes it.

8×34=68 \times \tfrac34 = 6, 8×1=88 \times 1 = 8, 8×54=108 \times \tfrac54 = 10. The factor below 11 shrank 88; the factor above 11 stretched it.

Example
Without calculating

Is 23×78\tfrac23 \times \tfrac78 more or less than 78\tfrac78? It is 23\tfrac23 of 78\tfrac78, and 23<1\tfrac23 < 1, so it is less. Check: 23×78=1424=712\tfrac23 \times \tfrac78 = \tfrac{14}{24} = \tfrac{7}{12}, and 712<78\tfrac{7}{12} < \tfrac78.

Example
Resizing a photo

A photo is 66 in wide. Enlarged by 1121\tfrac12: 6×32=96 \times \tfrac32 = 9 in. Shrunk to 23\tfrac23 of its size: 6×23=46 \times \tfrac23 = 4 in.

Side note
Where it goes wrong

Believing multiplication always makes things bigger. That is true when both factors are more than 11, but 10×12=510 \times \tfrac12 = 5. If one factor is less than 11, the product must be smaller than the other factor; an answer that is not smaller signals a mistake.

Try it
Try it: order the products

Without multiplying, put these in order from smallest to largest: 12×91012 \times \tfrac{9}{10}, 12×111012 \times \tfrac{11}{10}, 12×1212 \times \tfrac12. Then check by multiplying.

Show the answer

12\tfrac12 is the smallest factor and 1110\tfrac{11}{10} is the only one above 11, so the order is 12×1212 \times \tfrac12, then 12×91012 \times \tfrac{9}{10}, then 12×111012 \times \tfrac{11}{10}. Check: 6\mathbf{6}, 10.8\mathbf{10.8}, 13.2\mathbf{13.2}.

Why It Works

Draw a unit square. Shade 13\tfrac13 of it with vertical stripes and 12\tfrac12 of it with horizontal stripes. The doubly shaded part is 11 of the 66 equal rectangles the two cuts create, so 12×13=16\tfrac12\times\tfrac13 = \tfrac16. In general the cuts make b×db\times d equal pieces and the doubly shaded region has a×ca\times c of them: ab×cd=acbd\tfrac ab\times\tfrac cd = \tfrac{ac}{bd}.

Tip
Reading a proof

Do not read a proof the way you read a rule. Cover the last line, ask what the previous line forces, and only then uncover it. If a step feels like magic, that is the step to work by hand with a concrete number.