doing math

courses › Pre-Algebra

Ratios & proportions

level 19 course

Scale ratios, split amounts, and compare unit prices.

In your head, jot if needed · no calculator why?

Learn first (about 3 minutes)

Opens at level 19.

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Builds on: Fraction arithmetic (not open yet)

the lesson

The idea, the techniques and a tip for each skill, right here. The Learn page adds worked examples for every skill and untimed practice.

Read the lesson · about 3 minutes

The idea

A ratio like 3 : 4 compares two amounts: for every 3 of one, there are 4 of the other. Equal ratios come from multiplying both terms by the same number, never adding: 3 : 4 is the same as 6 : 8 and 21 : 28.

Prices and rates work the same way. Compare packs by the price per unit, and scale a rate from the amount for one unit.

Techniques

Find the scale factor

Two equal ratios with one term missing, like 3 : 4 = x : 28.

  1. Find the pair of matching terms that are both known, like 4 and 28.
  2. Divide the bigger by the smaller to get the scale factor: 28 ÷ 4 = 7.
  3. Scale the other term by the same factor: 3 × 7 = 21.
  4. If the missing term sits with the bigger numbers, multiply. With the smaller numbers, divide.
worked example

Example: 5 : 8 = x : 56. What is x?

  1. 56 ÷ 8 = 7, so the scale factor is 7.
  2. x = 5 × 7 = 35.

Answer: 35

Count the parts

Splitting an amount in a ratio, between two or three people.

  1. Add the ratio numbers to get the total number of parts.
  2. Divide the amount by the total parts to get one part.
  3. Multiply one part by the share you want.
worked example

Example: A bakery splits a $360 profit between two owners in the ratio 5 : 3. What is the smaller share?

  1. 5 + 3 = 8 parts.
  2. One part: $360 ÷ 8 = $45.
  3. The smaller share: 3 × $45 = $135.

Answer: $135

Price or amount per one

Comparing pack sizes, or scaling a rate to a new time.

  1. Divide to get the amount for one unit: dollars per ounce, pages per minute.
  2. Packs: the lowest price per unit wins, whatever the size.
  3. Rates: multiply the amount per unit by the new time.
worked example

Example: Coffee comes in two sizes: 12 oz for $3.60 and 20 oz for $5.40. What is the lower price per ounce?

  1. 12 oz: $3.60 ÷ 12 = $0.30 an ounce.
  2. 20 oz: $5.40 ÷ 20 = $0.27 an ounce.
  3. The 20 oz pack is the better value.

Answer: $0.27

Tips by skill

  • TipMissing term: Find the factor that turns one known term into its partner, then apply it to the other pair.
  • TipShare in a ratio: Add the ratio numbers for the total parts. Divide to get one part, then multiply by the share asked for.
  • TipBest buy: Divide each price by its size. The lowest price per unit is the best value, whatever the pack size.
  • TipScale a rate: Find the amount for one minute or one hour first, then multiply by the new time.

Watch out for

  • Adding the same amount to both terms. Equal ratios multiply: 3 : 4 scales to 21 : 28, not 27 : 28.
  • Dividing by the larger ratio number instead of the total parts. For 2 : 3, one part is the amount divided by 5.
  • Stopping at the rate per minute when the question asks about a longer time.

skills · practice stats

From rounds of this course only: box, review and test-out answers are left out. Once a skill has 40 tries, it compares your first 20 tries with your last 20.

rest ladder

Win 3 of your last 4 rounds and the course rests. A win is 90% right, within 2× the round's par. Pass the review when it comes back and the next rest is longer.

  1. 1 day
  2. 3 days
  3. 7 days
  4. 14 days
  5. 30 days
  6. 60 days
  7. mastered · every 90 days

your rounds

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