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Daily Math Minute

Ratios & Proportional Relationships

Unit Rates with Fractions

Computing unit rates associated with ratios of fractions.

Foundational15 min lesson3 min readUpdated August 12, 2026Author not yet attributed

Rates Where the Numbers Themselves Are Fractions

A runner covers 2/3 of a mile in 1/4 of an hour, holding a steady pace. Before reading on, think about how you'd find that runner's speed in plain miles per hour — a single, ordinary number, even though both quantities you're starting with are fractions.

A unit rate always means dividing a quantity by exactly 1 unit of another quantity — that part hasn't changed since Grade 6. What's new is that both the quantity and the unit itself can now be fractions, which means finding this unit rate is really a fraction division problem: 2/3 ÷ 1/4.

Worked Example — Finding a Unit Rate from Two Fractions

Find the runner's speed: 2/3 mile ÷ 1/4 hour. Divide by multiplying by the reciprocal: 2/3 × 4/1 = 8/3, which simplifies to 2 2/3. The runner's pace is 2 2/3 miles per hour.

Worked Example — Comparing Two Fractional Rates

One recipe uses 3/4 cup of sugar for every 1/2 cup of butter. Another uses 5/8 cup of sugar for every 1/3 cup of butter. Which recipe is sweeter, relative to its butter? Find each unit rate: (3/4) ÷ (1/2) = 3/2, and (5/8) ÷ (1/3) = 15/8. Since 15/8 (1 7/8) is greater than 3/2 (1 1/2), the second recipe has more sugar per cup of butter.
23÷14=23×41=83=223\dfrac{2}{3} \div \dfrac{1}{4} = \dfrac{2}{3} \times \dfrac{4}{1} = \dfrac{8}{3} = 2\tfrac{2}{3}

Tip

Set up a fractional unit rate exactly like any other rate — quantity divided by the amount of the other quantity — then reach for the reciprocal-multiplication skill from dividing fractions to actually carry it out.

Common Mistakes

  • Treating a rate involving fractions as too complicated to compute directly, and rounding the fractions to whole numbers before dividing.

    A fractional unit rate is found exactly the same way as any unit rate — divide the quantity by the other quantity — using fraction division rather than an estimate.

  • Setting up the division backward, dividing the smaller-context quantity by the larger one.

    Check which quantity the question wants reduced to '1 unit of' — that quantity always ends up as the divisor.

Key Takeaways

  • A unit rate with fractional quantities is found the same way as any unit rate, using fraction division.
  • Dividing by the reciprocal of the divisor turns a fraction-over-fraction rate into a single, comparable number.
  • Fractional unit rates can be compared directly once both are reduced to the same '1 unit' basis.

Summary

Fractional unit rates combine two skills you already have — unit rates and fraction division — into one calculation. The next lesson uses rates like these to decide whether two quantities are in a proportional relationship at all.

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