3.4 Fraction Multiplication as Scaling, Unit-Fraction Division, & Fractions in Measurement
Key Takeaways
- Multiplying by a fraction greater than 1 enlarges a quantity, multiplying by a fraction less than 1 shrinks it, and multiplying by exactly 1 in disguise (such as 4/4) leaves it unchanged.
- Because multiplying by a proper fraction makes a number smaller, "of" problems such as 2/3 of 45 are multiplication even though the answer is less than the original.
- Dividing a whole number by a unit fraction gives a larger result: 5 ÷ 1/4 = 20 because there are four quarters in every whole.
- Fraction arithmetic on TABE lives in measurement contexts — inch rulers, recipe scaling, and lumber cuts — so the unit must travel through every step of the calculation.
Fraction Multiplication as Scaling, Unit-Fraction Division, & Fractions in Measurement
Section 3.2 covered the procedures for fraction arithmetic. This section covers the reasoning TABE tests alongside them: predicting whether an answer should be bigger or smaller before you compute, and handling the measurement situations where Level M places most fraction items.
Multiplication as Scaling (Resizing)
Think of multiplication as resizing the first factor. The size of the second factor tells you which direction:
| Multiply by… | Effect on the product | Example |
|---|---|---|
| a number greater than 1 | product is larger than the original | $\tfrac{3}{2} \times 40 = 60$ |
| exactly 1 (including $\tfrac{5}{5}$, $\tfrac{12}{12}$) | product equals the original | $\tfrac{7}{7} \times 40 = 40$ |
| a number between 0 and 1 | product is smaller than the original | $\tfrac{3}{4} \times 40 = 30$ |
This is why generating an equivalent fraction works at all: multiplying $\frac{2}{3}$ by $\frac{4}{4}$ gives $\frac{8}{12}$, a different-looking fraction of exactly the same size, because $\frac{4}{4} = 1$.
Predict before you compute. If a TABE item asks for $\frac{5}{8}$ of 96 and one answer choice is 153.6, you can reject it instantly — $\frac{5}{8}$ is less than 1, so the answer must be less than 96. (It is 60.)
"Of" Means Multiply
The word of between a fraction and a quantity signals multiplication:
A faster route for mental math: divide by the denominator, multiply by the numerator. $45 \div 3 = 15$, then $15 \times 2 = 30$. This works every time and keeps the numbers small enough for Part 1.
Fraction of a fraction
Because both factors are less than 1, the product is smaller than either of them — which surprises learners the first time and is a favorite TABE distractor.
Dividing With Unit Fractions
A unit fraction has a numerator of 1. Division involving unit fractions runs in two directions, and mixing them up is the most common Level M fraction error.
Whole number ÷ unit fraction → bigger answer. The question is "how many of these small pieces fit?"
Five whole sandwiches cut into quarters make 20 quarter-sandwiches.
Unit fraction ÷ whole number → smaller answer. The question is "share this piece among this many."
A quarter of a pan shared among 5 people gives each one-twentieth of the pan.
| Situation | Setup | Result relative to start |
|---|---|---|
| How many $\frac{1}{3}$-cup scoops in 4 cups? | $4 \div \frac{1}{3}$ | 12 — larger |
| $\frac{1}{3}$ cup split among 4 bowls? | $\frac{1}{3} \div 4$ | $\frac{1}{12}$ cup — smaller |
Fractions in Measurement Contexts
Level M fraction items are rarely bare computation; they are rulers, recipes, and material.
Inch rulers and sixteenths
A standard inch ruler is marked in halves, quarters, eighths, and sixteenths. Reading it well means recognizing that $\frac{6}{16} = \frac{3}{8}$ and $\frac{12}{16} = \frac{3}{4}$ on sight.
A bolt measures $2\frac{5}{8}$ inches and passes through material $1\frac{3}{4}$ inches thick. How much bolt protrudes?
$2\frac{5}{8} - 1\frac{3}{4} = 2\frac{5}{8} - 1\frac{6}{8} = 1\frac{13}{8} - 1\frac{6}{8} = \mathbf{\frac{7}{8}}$ inch.
Recipe scaling
Scaling a recipe by $\frac{3}{2}$ (one and a half batches) or $\frac{1}{2}$ (half batch) multiplies every ingredient:
| Ingredient | Original | $\times \frac{3}{2}$ |
|---|---|---|
| flour | $2\frac{1}{4}$ cups | $\frac{9}{4} \times \frac{3}{2} = \frac{27}{8} = 3\frac{3}{8}$ cups |
| butter | $\frac{2}{3}$ cup | $\frac{2}{3} \times \frac{3}{2} = 1$ cup |
| salt | $\frac{1}{2}$ tsp | $\frac{3}{4}$ tsp |
Material cuts
A carpenter has a board $10\frac{1}{2}$ feet long and needs pieces $\frac{7}{8}$ foot long. How many complete pieces?
$10\frac{1}{2} \div \frac{7}{8} = \frac{21}{2} \times \frac{8}{7} = \frac{168}{14} = \mathbf{12}$ pieces.
Keep the unit attached at every step. "12" is meaningless; "12 pieces" is an answer.
Without computing the exact product, which statement is true about 7/9 × 342?
A cafeteria has 6 gallons of soup and serves it in 1/3-gallon bowls. How many bowls can be filled?
A pipe fitter cuts sections 3/4 foot long from a length of pipe measuring 8 1/4 feet. How many complete sections can be cut, assuming no waste?