3.4 Fraction Multiplication as Scaling, Unit-Fraction Division, & Fractions in Measurement

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:

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:

$$\frac{2}{3} \text{ of } 45 = \frac{2}{3} \times 45 = \frac{2 \times 45}{3} = \frac{90}{3} = \mathbf{30}$$

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

$$\frac{3}{4} \text{ of } \frac{2}{5} = \frac{3}{4} \times \frac{2}{5} = \frac{6}{20} = \frac{3}{10}$$

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?"

$$5 \div \frac{1}{4} = 5 \times 4 = \mathbf{20}$$

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."

$$\frac{1}{4} \div 5 = \frac{1}{4} \times \frac{1}{5} = \mathbf{\frac{1}{20}}$$

A quarter of a pan shared among 5 people gives each one-twentieth of the pan.

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:

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.