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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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  1. MINI QUEST 4.1Equivalent Fractions & ComparisonRead the lesson
  2. MINI QUEST 4.2Combining & Scaling FractionsRead the lesson
  3. 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.

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.

a/b means a copies of 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.

a/b = ak/bk, b ≠ 0, k ≠ 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.

a/b < c/d ⇔ ad < bc for b,d > 0
WORKED EXAMPLE 1

Reduce 42/70.

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

3/5

WORKED EXAMPLE 2

Which is larger, 5/8 or 7/12?

  1. Use denominator 24: the fractions are 15/24 and 14/24.
  2. Compare the equal-sized parts.

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
  1. Reduce 36/60 to a fraction.
  2. Find n if n/35 = 4/7.
  3. Which fraction is larger: 7/10 or 2/3? Enter the larger fraction.
  4. Which is smaller: −3/4 or −2/3? Enter the smaller fraction.
  5. What numerator makes 5/6 = n/42?
  6. How many integers n satisfy 1/3 < n/12 < 3/4?
  7. Two identical tanks contain 5/9 and 7/12 of their capacities. Which fraction describes the fuller tank? New context
  8. 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? New context
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.

a/b + c/b = (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.

(a/b)(c/d) = ac/bd

Division counts groups

Dividing by c/d asks how many groups of that size fit. Multiply by d/c when c is nonzero. Check by multiplying the quotient by the divisor.

(a/b) ÷ (c/d) = ad/bc
WORKED EXAMPLE 1

Compute 5/6 − 3/8.

  1. Use denominator 24: 20/24 − 9/24.
  2. Subtract numerators and keep denominator 24.

11/24

WORKED EXAMPLE 2

A 7/8-meter cord is cut into 1/16-meter pieces. How many pieces?

  1. Divide total length by piece length.
  2. (7/8) × 16 = 14.

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
  1. Compute 2/3 + 5/12.
  2. Compute 7/10 − 1/4.
  3. Compute (9/14)(7/15).
  4. Compute (5/6) ÷ (10/9).
  5. Compute 1 − (2/5 + 1/6).
  6. Find x if x + 3/7 = 5/6.
  7. A recipe needs 3/4 cup of oats per batch. How many cups are needed for 2/3 of a batch? New context
  8. A jug holds 7/3 liters. After 5/6 liter is used, the rest fills identical 1/4-liter cups. How many cups can be filled exactly? New context
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.

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.

(a/b)ⁿ = aⁿ/bⁿ

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.

0 < r < 1 ⇒ r² < r
WORKED EXAMPLE 1

Compute (1 2/3)(2 1/4).

  1. Convert to 5/3 and 9/4.
  2. Multiply and reduce 45/12.

15/4

WORKED EXAMPLE 2

Compute (−2/5)³.

  1. Cube both parts and preserve the odd-power negative sign.
  2. The magnitude is 8/125.

−8/125

Common pitfalls

Possible mix-up: 1 2/3 means 1 × 2/3.

Mixed notation means 1 + 2/3.

Possible mix-up: (2/3)² equals 4/3.

The denominator is squared too, giving 4/9.

Explain it to yourself

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

Preview the eight practice prompts
  1. Write 3 2/5 as a single fraction.
  2. Compute 1 3/4 + 2 2/3. Enter a fraction or decimal.
  3. Compute (2/5)³.
  4. Compute (−3/4)².
  5. Compute (2 1/2) ÷ (1 1/4).
  6. Compute 1 − (3/5)².
  7. A square tile has side length 1 1/2 centimeters. What is its area in square centimeters? New context
  8. A 6 3/4-meter strip is divided into equal pieces of length 1 1/8 meters. How many pieces result? New context
Open stage 4.3 in the student workspace →
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