Ratios, Rates, and Proportions

Ratio, rate, proportion: what each word means

• Ratio, a comparison of two quantities with the same unit. "3 girls to 2 boys" is written 3 : 2, or as the fraction 3⁄2.
• Rate, a comparison of two quantities with different units, like kilometres and hours (km/h) or dollars and litres ($/L).
• Unit rate, a rate written "per 1" of something: 100 km in 2 hours becomes 50 km per 1 hour.
• Proportion, an equation saying two ratios are equal, like 3⁄2 = 9⁄6. We solve these to find a missing number.

Worked example #1: writing and simplifying a ratio

A class has 18 students who walk to school and 12 who take the bus. Write the ratio of walkers to bus riders in simplest form.

• Write it as given: walkers to riders is 18 : 12, or 18⁄12.
• Find a common factor: both 18 and 12 divide by 6.
• Divide both sides: 18 ÷ 6 = 3 and 12 ÷ 6 = 2.

  18 : 12
= (18 ÷ 6) : (12 ÷ 6)
= 3 : 2

Worked example #2: unit rate (which is the better buy?)

One store sells a 4 L jug of milk for $6.40. Another sells a 2 L carton for $3.50. Which is the better deal? Find the price per litre for each, that is the unit rate.

• Set up each rate: dollars on top, litres on the bottom, then divide to get the price for 1 litre.
• 4 L jug: $6.40 ÷ 4 = $1.60 per litre.
• 2 L carton: $3.50 ÷ 2 = $1.75 per litre.
• Compare: $1.60 is less than $1.75 per litre.

  4 L jug:    $6.40 ÷ 4 = $1.60 per litre
  2 L carton: $3.50 ÷ 2 = $1.75 per litre

  $1.60 < $1.75  →  the 4 L jug is cheaper

The key move: solving a proportion by cross-multiplying

A proportion is two equal ratios with one number missing, usually called x. To find it, cross-multiply: multiply each top by the opposite bottom, set the two products equal, then divide to get x on its own.

The dashed lines show the two cross-products: 4 × x and 3 × 20. Set them equal (4x = 60), then divide both sides by 4 to get x = 15.

Worked example #3: scaling a recipe with a proportion

A muffin recipe uses 3 cups of flour to make 12 muffins. You want to make 30 muffins. How much flour do you need? This is a perfect proportion: set up the two ratios the same way, flour on top, muffins on the bottom, on both sides.

• Set up the proportion: 3 cups is to 12 muffins as x cups is to 30 muffins.
• Cross-multiply: 12 × x = 3 × 30, so 12x = 90.
• Solve for x: 90 ÷ 12 = 7.5.

Worked example #4: a map scale problem

On a map, 2 cm represents 5 km of real distance. Two towns are 14 cm apart on the map. How far apart are they in real life? Set up map-distance over real-distance on both sides.

• Set up the proportion: 2 cm is to 5 km as 14 cm is to x km.
• Cross-multiply: 2 × x = 5 × 14, so 2x = 70.
• Solve for x: 70 ÷ 2 = 35.

Speed is just a rate (km/h)

The formula sheet gives you distance = speed × time, but it is really a rate problem. Speed in km/h is a unit rate: kilometres per 1 hour. If a car travels 240 km in 3 hours, divide to get the unit rate.

  speed = 240 km ÷ 3 h = 80 km/h

You can also run it as a proportion: 240 km is to 3 h as x km is to 1 h. Cross-multiplying gives 3x = 240, so x = 80 km/h. Same answer, same idea, pick whichever setup feels clearer to you.

Tips that keep proportions tidy

• Label your fractions. Write the units beside the numbers (flour/muffins, cm/km). It instantly tells you whether both sides match.
• Keep the same quantity on top on both sides. That is the single most common setup mistake, flipping one side upside down gives a wrong answer.
• Cross-multiply, then divide. Multiply across the diagonal, set the products equal, then divide both sides by the number multiplying x.
• Sanity-check the size. If you scaled up, the answer should be bigger; if you scaled down, smaller. A quick glance catches calculator slips.

Your turn: practice problems

Set each one up carefully, write the units, keep the same quantity on top, then cross-multiply. A calculator is fine. No peeking until you have tried.

1. Simplify the ratio 20 : 8 to simplest form.
2. A 750 mL bottle of juice costs $2.40, and a 1 L bottle costs $3.40. Which has the lower price per millilitre?
3. A recipe uses 2 cups of rice for 5 servings. How many cups are needed for 8 servings?
4. On a map, 3 cm represents 12 km. Two cities are 7 cm apart on the map. How far apart are they in real life?