Math Vale

🍰 Fraction Isles · Fractions & Proportion

Ratios & Proportions

Compare quantities with ratios, find unit rates, and solve proportions that scale a relationship up or down.

In short

  • Equivalent ratios are made by multiplying or dividing both parts by the same number, never by adding to them.
  • A unit rate reduces one quantity to 1, which is the only fair way to compare offers of different sizes.
  • Cross-multiplying pairs each numerator with the denominator diagonally opposite it.
  • To share a total in a ratio, divide by the sum of the parts to find one share, then multiply.
  • A part-to-part ratio of 3 : 5 makes the first group 3/8 of the whole, not 3/5.

A ratio compares, a fraction divides

A ratio compares two quantities of the same kind. Writing "archers to shield-bearers is 3 : 5" says that for every 3 archers there are 5 shield-bearers — it does not say there are exactly 3 and 5.

Ratios behave like fractions: multiply or divide both parts by the same number and the comparison is unchanged.

  • 3 : 5 = 6 : 10 = 30 : 50 = 300 : 500

What you may never do is add the same number to both parts. 3 : 5 is not 4 : 6, because 4/6 is not 3/5. Ratios scale by multiplying, never by adding.

Careful with the difference between a part-to-part ratio and a part-to-whole fraction. If the ratio is 3 : 5 then there are 8 shares in total, so archers are 3/8 of the company, not 3/5.

Unit rates: what one costs

A rate compares two quantities of different kinds — gold per day, kilometres per hour, arrows per quiver. A unit rate rewrites it so the second quantity is exactly 1, and that is what makes offers comparable.

Two bundles of arrows cannot be compared by price alone if they hold different numbers of arrows. Divide the price by the count in each case:

  • 14 arrows for 84 gold: 84 / 14 = 6 gold each
  • 9 arrows for 63 gold: 63 / 9 = 7 gold each

The first bundle is better value even though it costs more in total. Cheapest total and best value are different questions.

A table makes the divisions visible and is worth drawing:

  • Bundle | Arrows | Gold | Gold each

Proportions and cross-multiplying

A proportion is a statement that two ratios are equal, such as 3/5 = x/20.

There are two good ways to solve one.

Scale factor. Look across for the number that turns one fraction into the other. Here 5 becomes 20, so the factor is 4; apply it to the top as well and x = 12. This is the quicker method whenever the factor is a whole number.

Cross-multiplying. Multiply each numerator by the denominator diagonally opposite:

  • 3/5 = x/20 gives 3 x 20 = 5 x x, so 60 = 5x and x = 12.

Cross-multiplying always works, including when the scale factor is awkward. The one thing to be careful about is which numbers pair up: a numerator pairs with the opposite denominator, never with the one below it.

Sharing a total in a ratio

When a total is split in a given ratio, count the shares first.

Suppose 96 supplies are split between two camps in the ratio 5 : 3.

  • Total shares = 5 + 3 = 8
  • One share = 96 / 8 = 12
  • The camps get 5 x 12 = 60 and 3 x 12 = 36
  • Check: 60 + 36 = 96

The mistake to avoid is dividing the total by one part of the ratio instead of by the sum of the parts. Always ask "how many equal shares is the whole being cut into?" before dividing.

Scaling recipes, maps and models

Proportional reasoning is what lets you resize something without distorting it.

Recipes. A stew for 4 becomes a stew for 12 by multiplying *every* ingredient by 3. Adding 8 to each ingredient would ruin it, because the scale factor is 12/4 = 3, not the difference 8.

Maps. A scale of "2 cm to 1 km" is a rate. Going from ground to map you multiply by 2; going from map to ground you divide by 2. Setting it out as a proportion keeps the direction straight:

  • 2 cm / 1 km = 9 cm / x km, so 2x = 9 and x = 4.5 km.

Similar shapes. Enlarging a rectangle proportionally means multiplying both the length and the width by the same factor — which is why doubling the sides of a square makes its area four times bigger, not twice.

Worked examples

Example 1

Solve the proportion 4/7 = x/56.

  1. Look across the equals sign: the denominator 7 has become 56.
  2. 56 / 7 = 8, so the scale factor is 8.
  3. Apply the same factor to the numerator: x = 4 x 8 = 32.
  4. Cross-multiplying gives the same answer: 4 x 56 = 7x, so 224 = 7x and x = 32.
  5. Check: 32/56 divides down by 8 to 4/7.

Example 2

A guild splits 132 gold between a scout and a healer in the ratio 7 : 5. How much does the healer receive?

  1. Count the shares: 7 + 5 = 12 equal shares in total.
  2. Find one share: 132 / 12 = 11 gold.
  3. The healer takes 5 shares: 5 x 11 = 55 gold.
  4. Check the other part too: the scout takes 7 x 11 = 77 gold, and 55 + 77 = 132.

Practice problems, with solutions

Three problems of increasing difficulty, each with the full working. In the game these are generated fresh every time; these three are fixed so this page always shows the same ones.

Problem 1

Difficulty 1 of 5

These two ratios are equivalent. Find the missing number. 4 : 3 = 16 : ?

Answer: 12

  1. Compare the first parts: 4 x ? = 16, so the scale factor is 4.
  2. Apply the same factor to the second part: 3 x 4 = 12.
  3. Check: 16 : 12 simplifies back to 4 : 3.

Problem 2

Difficulty 3 of 5

Ravi records 87 arrows across 3 quivers. What is the rate per quiver?

Answer: 29 arrows per quiver

  1. Rate = total / number of quivers
  2. = 87 / 3
  3. = 29 arrows per quiver
  4. Check by scaling back up: 29 x 3 = 87.

Problem 3

Difficulty 4 of 5

A quartermaster offers ink pots in bundles. Which bundle is the better value?

  1. 5 ink pots for 70 gold
  2. 9 ink pots for 135 gold
  3. 4 ink pots for 48 gold

Answer: C. 4 ink pots for 48 gold

  1. 4 ink pots for 48 gold: 48 / 4 = 12 gold each.
  2. 5 ink pots for 70 gold: 70 / 5 = 14 gold each.
  3. 9 ink pots for 135 gold: 135 / 9 = 15 gold each.
  4. The lowest cost per item is 12 gold, so "4 ink pots for 48 gold" is the better value.

Common mistakes

  • Adding the same number to both parts of a ratio instead of scaling them.
  • Cross-multiplying the wrong pair, so a numerator is matched with the denominator directly below it.
  • Choosing the cheapest total price rather than the lowest cost per item.
  • Dividing a total by one part of the ratio instead of by the sum of the parts.
  • Confusing a part-to-part ratio with a part-to-whole fraction.

What you should be able to do

  • Write and simplify a ratio, and find an equivalent ratio.
  • Compute a unit rate and use it to compare deals.
  • Solve a proportion for an unknown value.
  • Use proportional reasoning for scaling, recipes and maps.

Where this fits in the curriculum

Common Core

  • 6.RP.A.1

    Grade 6 — Understand the concept of a ratio and describe a relationship between two quantities with it.

  • 6.RP.A.2

    Grade 6 — Understand a unit rate and use rate language.

  • 7.RP.A.2

    Grade 7 — Recognise and represent proportional relationships, and find the constant of proportionality.

Ontario

  • G6.B2.12

    Grade 6 — Solve problems involving ratios, rates and percents.

  • G7.B2.10

    Grade 7 — Identify proportional and non-proportional situations and apply proportional reasoning to solve problems.

  • G8.B2.8

    Grade 8 — Solve problems involving rates, ratios and proportions in various contexts.

  • MTH1W.B3.5

    Grade 9 de-streamed — Solve problems involving rates, percentages and proportions in various contexts.

SAT

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