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Prealgebra / LEVEL 4 · DIFFICULTY 2/5

Fractions & Equal Shares

Compare, combine and reinterpret quantities measured in equal parts.

3 stages · 24 practice problems · two 6-question assessment forms

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MINI QUEST 4.1Equivalent Fractions & ComparisonRead the lesson
MINI QUEST 4.2Combining & Scaling FractionsRead the lesson
MINI QUEST 4.3Mixed Numbers & Fraction PowersRead the lesson
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STAGE 4.1 · 8 PRACTICE PROBLEMS · 2 NEW-CONTEXT APPLICATIONS

Equivalent Fractions & Comparison

Useful preparation: Greatest Common Divisor

Goal: Preserve a fraction value while changing its representation.

Before you begin: Before starting, explain one example from each prerequisite. If an idea is unfamiliar, follow the prerequisite lesson links.

MOVE FIRST. WRITE THE MATH AFTER.

Share 12 counters into 3 equal groups

Drag a counter into a group. Or tap a counter, then “Put selected here.” Everyone should get the same number.

One small move: Move one counter from the tray into group 1. It has 0 right now.

Counter tray · 12 left

Group $\displaystyle 1 \cdot 0$

Group $\displaystyle 2 \cdot 0$

Group $\displaystyle 3 \cdot 0$

Move one counter from the tray into group 1. It has 0 right now.

Keep the whole fixed

A denominator tells how many equal parts form one whole. The numerator counts those parts. Comparisons require the same size whole.

\displaystyle \frac{a}{b}

means a copies of

\displaystyle \frac{1}{b}

Scale numerator and denominator together

Multiplying both by the same nonzero number changes the part size and part count in compensating ways. Reduce by dividing both by a common divisor.

\displaystyle \frac{a}{b}

= ak/bk,

\displaystyle b \ne 0

\displaystyle k \ne 0

Compare using equal units

Use common denominators or compare cross-products when denominators are positive. Benchmark fractions such as one half can save work.

\displaystyle \frac{a}{b} < \frac{c}{d}

⇔

\displaystyle ad < bc

for b,

\displaystyle d > 0

WORKED EXAMPLE 1

Reduce $\displaystyle \frac{42}{70}$ .

The greatest common divisor is 14.
Divide numerator and denominator by 14.

$\displaystyle \frac{3}{5}$

WORKED EXAMPLE 2

Which is larger, $\displaystyle \frac{5}{8}$ or $\displaystyle \frac{7}{12}$ ?

Use denominator 24: the fractions are $\displaystyle \frac{15}{24}$ and $\displaystyle \frac{14}{24}$ .
Compare the equal-sized parts.

$\displaystyle \frac{5}{8}$

Common pitfalls

Possible mix-up: A larger denominator always means a larger fraction.

For equal positive numerators, more parts per whole means smaller parts.

Possible mix-up: Add 2 to the top and bottom to get an equivalent fraction.

Adding the same number usually changes the ratio; multiply or divide both instead.

Explain it to yourself

Why must cross-product comparison account for the signs of denominators?

Preview the eight practice prompts

Problem 1
Reduce $\displaystyle \frac{36}{60}$ to a fraction.
Practise this exact problem →
Problem 2
Find n if $\displaystyle \frac{n}{35} = \frac{4}{7}$ .
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Problem 3
Which fraction is larger: $\displaystyle \frac{7}{10}$ or $\displaystyle \frac{2}{3}$ ? Enter the larger fraction.
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Problem 4
Which is smaller: $\displaystyle \frac{-3}{4}$ or $\displaystyle \frac{-2}{3}$ ? Enter the smaller fraction.
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Problem 5
What numerator makes $\displaystyle \frac{5}{6} = \frac{n}{42}$ ?
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Problem 6
How many integers n satisfy $\displaystyle \frac{1}{3} < \frac{n}{12} < \frac{3}{4}$ ?
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Problem 7New context
Two identical tanks contain $\displaystyle \frac{5}{9}$ and $\displaystyle \frac{7}{12}$ of their capacities. Which fraction describes the fuller tank?
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Problem 8New context
A design uses 18 blue squares among 30 total. A larger version keeps that fraction and has 45 total squares. How many must be blue?
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Open stage 4.1 in the student workspace →

STAGE 4.2 · 8 PRACTICE PROBLEMS · 2 NEW-CONTEXT APPLICATIONS

Combining & Scaling Fractions

Useful preparation: Equivalent Fractions & Comparison · Division, Reciprocals & Zero

Goal: Choose common units for sums and reciprocal factors for division.

Before you begin: Before starting, explain one example from each prerequisite. If an idea is unfamiliar, follow the prerequisite lesson links.

Addition needs equal-sized parts

To add or subtract fractions, first express them using a common denominator. Add or subtract the numerators and retain the common part size.

\displaystyle \frac{a}{b} + \frac{c}{b} = \frac{a + c}{b}

Multiplication scales a quantity

A fraction of a fraction multiplies the scale factors. Cancel common factors across the product before multiplying when convenient.

\displaystyle \left(\frac{a}{b}\right)\, \left(\frac{c}{d}\right) = \frac{ac}{bd}

Division counts groups

Dividing by $\displaystyle \frac{c}{d}$ asks how many groups of that size fit. Multiply by $\displaystyle \frac{d}{c}$ when c is nonzero. Check by multiplying the quotient by the divisor.

\displaystyle \left(\frac{a}{b}\right) \div \left(\frac{c}{d}\right) = \frac{ad}{bc}

WORKED EXAMPLE 1

Compute $\displaystyle \frac{5}{6} - \frac{3}{8}$ .

Use denominator $\displaystyle 24 : \frac{20}{24} - \frac{9}{24}$ .
Subtract numerators and keep denominator 24.

$\displaystyle \frac{11}{24}$

WORKED EXAMPLE 2

$\displaystyle \frac{A\, 7}{8}$ -meter cord is cut into $\displaystyle \frac{1}{16}$ -meter pieces. How many pieces?

Divide total length by piece length.
$\displaystyle \left(\frac{7}{8}\right) \times 16 = 14$ .

Common pitfalls

Possible mix-up: Add denominators when adding fractions.

Denominators name the parts; first convert to the same part size.

Possible mix-up: Cancel a term across a sum.

Cancellation removes common factors, not individual addends.

Explain it to yourself

When can a fraction quotient be larger than the dividend?

Preview the eight practice prompts

Problem 1
Compute $\displaystyle \frac{2}{3} + \frac{5}{12}$ .
Practise this exact problem →
Problem 2
Compute $\displaystyle \frac{7}{10} - \frac{1}{4}$ .
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Problem 3
Compute $\displaystyle \left(\frac{9}{14}\right)\, \left(\frac{7}{15}\right)$ .
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Problem 4
Compute $\displaystyle \left(\frac{5}{6}\right) \div \left(\frac{10}{9}\right)$ .
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Problem 5
Compute $\displaystyle 1 - \left(\frac{2}{5} + \frac{1}{6}\right)$ .
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Problem 6
Find x if $\displaystyle x + \frac{3}{7} = \frac{5}{6}$ .
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Problem 7New context
A recipe needs $\displaystyle \frac{3}{4}$ cup of oats per batch. How many cups are needed for $\displaystyle \frac{2}{3}$ of a batch?
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Problem 8New context
A jug holds $\displaystyle \frac{7}{3}$ liters. After $\displaystyle \frac{5}{6}$ liter is used, the rest fills identical $\displaystyle \frac{1}{4}$ -liter cups. How many cups can be filled exactly?
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Open stage 4.2 in the student workspace →

STAGE 4.3 · 8 PRACTICE PROBLEMS · 2 NEW-CONTEXT APPLICATIONS

Mixed Numbers & Fraction Powers

Useful preparation: Combining & Scaling Fractions · Powers, Signs & Grouping

Goal: Convert mixed notation and apply powers to entire fractions.

Before you begin: Before starting, explain one example from each prerequisite. If an idea is unfamiliar, follow the prerequisite lesson links.

A mixed number is a sum

Convert a whole number plus a proper fraction into one fraction before multiplying or dividing. The denominator stays fixed.

\displaystyle \frac{w\, a}{b}

= (wb + a)/b

Power both numerator and denominator

Repeated multiplication of a fraction raises both parts to the same power. Parentheses identify exactly what is being raised.

\displaystyle \left(\frac{a}{b}\right)^{n} = \frac{a^{n}}{b^{n}}

Check the scale of the result

Squaring a positive fraction below one makes it smaller. Squaring a number above one makes it larger. Negatives require attention to parity.

\displaystyle 0 < r < 1 \Rightarrow r^{2} < r

WORKED EXAMPLE 1

Compute $\displaystyle \left(\frac{1\, 2}{3}\right)\, \left(\frac{2\, 1}{4}\right)$ .

Convert to $\displaystyle \frac{5}{3}$ and $\displaystyle \frac{9}{4}$ .
Multiply and reduce $\displaystyle \frac{45}{12}$ .

$\displaystyle \frac{15}{4}$

WORKED EXAMPLE 2

Compute $\displaystyle \left(\frac{-2}{5}\right)^{3}$ .

Cube both parts and preserve the odd-power negative sign.
The magnitude is $\displaystyle \frac{8}{125}$ .

$\displaystyle \frac{-8}{125}$

Common pitfalls

Possible mix-up: $\displaystyle \frac{1\, 2}{3}$ means $\displaystyle \frac{1 \times 2}{3}$ .

Mixed notation means $\displaystyle 1 + \frac{2}{3}$ .

Possible mix-up: $\displaystyle \left(\frac{2}{3}\right)^{2}$ equals $\displaystyle \frac{4}{3}$ .

The denominator is squared too, giving $\displaystyle \frac{4}{9}$ .

Explain it to yourself

How does squaring change a positive number on each side of 1?

Preview the eight practice prompts

Problem 1
Write $\displaystyle \frac{3\, 2}{5}$ as a single fraction.
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Problem 2
Compute $\displaystyle \frac{1\, 3}{4} + \frac{2\, 2}{3}$ . Enter a fraction or decimal.
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Problem 3
Compute $\displaystyle \left(\frac{2}{5}\right)^{3}$ .
Practise this exact problem →
Problem 4
Compute $\displaystyle \left(\frac{-3}{4}\right)^{2}$ .
Practise this exact problem →
Problem 5
Compute $\displaystyle \left(\frac{2\, 1}{2}\right) \div \left(\frac{1\, 1}{4}\right)$ .
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Problem 6
Compute $\displaystyle 1 - \left(\frac{3}{5}\right)^{2}$ .
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Problem 7New context
A square tile has side length $\displaystyle \frac{1\, 1}{2}$ centimeters. What is its area in square centimeters?
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Problem 8New context
$\displaystyle \frac{A\, 6\, 3}{4}$ -meter strip is divided into equal pieces of length $\displaystyle \frac{1\, 1}{8}$ meters. How many pieces result?
Practise this exact problem →

Open stage 4.3 in the student workspace →

Browse another level →

Fractions & Equal Shares

Equivalent Fractions & Comparison

Share 12 counters into 3 equal groups

Keep the whole fixed

Scale numerator and denominator together

Compare using equal units

Reduce 4270\displaystyle \frac{42}{70}7042​.

Which is larger, 58\displaystyle \frac{5}{8}85​ or 712\displaystyle \frac{7}{12}127​?

Common pitfalls

Explain it to yourself

Combining & Scaling Fractions

Addition needs equal-sized parts

Multiplication scales a quantity

Division counts groups

Compute 56−38\displaystyle \frac{5}{6} - \frac{3}{8}65​−83​.

A 78\displaystyle \frac{A\, 7}{8}8A7​-meter cord is cut into 116\displaystyle \frac{1}{16}161​-meter pieces. How many pieces?

Common pitfalls

Explain it to yourself

Mixed Numbers & Fraction Powers

A mixed number is a sum

Power both numerator and denominator

Check the scale of the result

Compute (1 23) (2 14)\displaystyle \left(\frac{1\, 2}{3}\right)\, \left(\frac{2\, 1}{4}\right)(312​)(421​).

Compute (−25)3\displaystyle \left(\frac{-2}{5}\right)^{3}(5−2​)3.

Common pitfalls

Explain it to yourself

Reduce $\displaystyle \frac{42}{70}$ .

Which is larger, $\displaystyle \frac{5}{8}$ or $\displaystyle \frac{7}{12}$ ?

Compute $\displaystyle \frac{5}{6} - \frac{3}{8}$ .

$\displaystyle \frac{A\, 7}{8}$ -meter cord is cut into $\displaystyle \frac{1}{16}$ -meter pieces. How many pieces?

Compute $\displaystyle \left(\frac{1\, 2}{3}\right)\, \left(\frac{2\, 1}{4}\right)$ .

Compute $\displaystyle \left(\frac{-2}{5}\right)^{3}$ .