SAT Math · Problem-Solving and Data Analysis · Skill 1 of 7

Ratios, rates and proportions

Scale a ratio, find the better buy, solve any proportion and convert units so they cancel cleanly. These ideas sit behind most SAT word problems.

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Hi, I’m Io. If you can scale a recipe, you can do this whole lesson.
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Mix it to the recipe

Explore · 5 min

A lemonade recipe uses 2 cups of syrup for every 5 cups of water, a ratio of 2 : 5. To make more, multiply both parts by the same number. Three times the recipe is 6 : 15, and it tastes exactly the same.

A ratio compares part to part. In 2 : 5 there are 2 + 5 = 7 cups in each batch, so syrup is 2/7 of the whole mix. Many SAT questions turn on that difference.

Unit rates and the better buy

Learn · 4 min

A rate compares two different quantities, such as miles and gallons. The unit rate is the amount for one: divide the first quantity by the second. Once you have the unit rate, any other amount is one multiplication away.

unit rate = amount ÷ number of units

Solving proportions

Practise · 5 min

A proportion says two ratios are equal. Set it up with matching units in matching positions, then cross multiply: if a/b = c/d, then a × d = b × c.

Worked example: 3 notebooks cost $8. How much do 21 notebooks cost?
  1. Notebooks over dollars on both sides: 3/8 = 21/x
  2. Cross multiply: 3x = 8 × 21 = 168
  3. Divide by 3: x = 56, so 21 notebooks cost $56
  4. Check: 21 is 7 times 3, and 56 is 7 times 8

Proportional relationships

Game · 4 min

Two quantities are proportional when y = kx for some constant k. Three quick tests:

  • The ratio y/x is the same for every pair of values
  • The graph is a straight line through the origin
  • There is no starting amount: 0 of one means 0 of the other
Sort each relationship

Build a unit chain

Explore · 5 min

To convert units, multiply by fractions that are each equal to 1, such as 5280 feet / 1 mile. Choose the version that puts the old unit on the opposite side, so it cancels. Tap conversion cards to build each chain.

Square units, cubic units and density

Learn · 4 min

1 foot is 12 inches, but a square foot is 12 inches by 12 inches, so 1 square foot = 144 square inches. For cubic units, cube the factor: 1 cubic yard = 3 × 3 × 3 = 27 cubic feet.

Density is a rate too: mass per unit of volume. Rearrange it to find whichever quantity is missing.

density = mass ÷ volume

Common slipConverting 3 square yards by multiplying by 3 gives 9. The factor for square units is 3² = 9, so the answer is 27 square feet.

Real-life rates

Scenarios · 6 min

Maps, fuel and recipes are the SAT’s favourite settings for rates. Pick a scenario and answer each question.

Rates in two steps

Practise · 4 min

Harder rate questions chain two ideas. For motion, use distance = rate × time. When two machines or pumps work together, add their rates.

Worked example: Machine A makes 120 parts an hour and machine B makes 80. How long do they take together to make 1,000 parts?
  1. Combined rate: 120 + 80 = 200 parts per hour
  2. Time = amount ÷ rate = 1,000 ÷ 200
  3. So together they take 5 hours

Spot the trap

Detective · 3 min

Each solution has one faulty step, and each one matches a wrong answer choice you will see on test day.

Final challenge

8 questions · 10 min

Key ideas to carry in
ScalingMultiply both parts of a ratio by the same number.
Part or wholeIn a : b, the first part is a/(a + b) of the whole.
Unit rateDivide to get the amount for one, then multiply.
ProportionsMatch units in matching positions, then cross multiply.
Unit chainsPut the unit to cancel on the opposite side.
Square unitsSquare the factor. Cubic units: cube it.
Challenge