Ratio, Rates and Proportion

Ratio is the first topic where the arithmetic is easy and the reading is hard. Almost nobody gets 360 ÷ 9 wrong; plenty of people divide by 4 when the ratio was 4:5, because they compared a part with a part where the question compared a part with the whole.

So this page is organised around that distinction. A ratio compares two parts with each other. A fraction compares one part with the whole. A rate compares two quantities in different units. Get those three straight and the rest is multiplication.

What is a ratio?

A ratio says how the parts of something relate in size. Ratio starts with the colon notation — red : blue = 2 : 3 means two reds for every three blues — and Part-to-Whole Ratios immediately does the conversion that everything else depends on: adding the parts gives five, so red is $\frac{2}{5}$2/5 of the total, not $\frac{2}{3}$2/3.

Tidying a Ratio works like simplifying a fraction — divide every part by their common factor, so 12 : 18 becomes 2 : 3 — and it extends to Three-Part Ratios without changing. Ratio, Fraction, Percent completes the translation table: 2 : 3 is $\frac{2}{5}$2/5 of the whole, which is 40%.

Sharing an amount in a ratio

Add the parts, divide, multiply out. Share 360 in the ratio 4 : $5\to 9$5 → 9 parts → one part is 40 → shares of 160 and 200. The check is that the shares add back to the original.

Bar Models for Ratio is the drawing that makes this hard to get wrong: nine equal boxes in two groups, and the question becomes obvious. It stays useful far longer than it looks — the trickiest problems on this page are still solvable by drawing the bars.

Combining Two Ratios handles the case with a shared middle term. If A : B = 2 : 3 and B : C = 4 : 5, the two Bs disagree, so scale each ratio until they match: A : B = 8 : 12 and B : C = 12 : 15, giving A : B : C = 8 : 12 : 15.

The Unitary Method is the general-purpose tool underneath all of this: find one unit, then multiply up. 7 notebooks cost 21, so one costs 3, so 12 cost 36. It is never the fastest method and it never fails, which makes it the right thing to reach for when a question is worded to confuse. Scale Factor is the same idea named as a multiplier — the number that takes one quantity to its partner.

What is a rate?

A rate compares two quantities measured in different units, so its value carries both. Rates introduces the idea and Rate, Total, Units gives the triangle that connects them: rate × units = total, and any two of the three find the third.

Comparing Unit Rates is the practical skill — a 750 g box at 3.60 against a 1.2 kg box at 5.40 is 0.48 against 0.45 per 100 g — and it is the whole content of a supermarket shelf label. Exchange Rates is the same arithmetic with the added care of knowing which way the rate points, which is the only genuinely error-prone part.

Speed, density and pressure

Speed is distance per unit time, the rate everyone already owns intuitively. Average Speed is where intuition fails: it is total distance divided by total time, not the average of the speeds. Driving out at 60 and back at 30 over the same road averages 40, not 45, because the slow leg takes twice as long and therefore counts twice as much.

Density (mass per volume) and Pressure (force per area) are the same triangle with different labels, which is the point of teaching them together — one relationship, three subjects.

Converting Compound Units is the step that gets skipped in physics as often as in mathematics. Convert the top and the bottom separately: 72 km/h → 72,000 m/h → 72,000 ÷ 3,600 = 20 m/s. Handling both at once is where the errors live.

Direct proportion and inverse proportion

In direct proportion, doubling one quantity doubles the other and the ratio stays fixed. In inverse proportion, doubling one halves the other and the product stays fixed.

Direct Proportion is y = kx, a straight line through the origin. Inverse Proportion is xy = k: four painters taking six days means eight painters take three, because the total work — twenty-four painter-days — is what stays constant.

The Constant of Proportionality is the number k itself, and naming it is what turns a proportion question into an equation you can solve rather than a pattern you have to spot. Proportion by Unit Value is the unitary method applied here, one step at a time.

Map Scales is direct proportion with a fixed constant, and the notation is worth pinning down: 1 : 50,000 means 1 cm on the map is 50,000 cm — half a kilometre — on the ground.

Before-and-after ratio problems

These are the questions that separate confident students from the rest, and the method is always the same: find what did not change, and scale both ratios so that the unchanged thing is written with the same number of parts.

Rewriting a Ratio to Match a Share teaches the scaling move itself. Then there are three shapes, and they cover almost every exam question of this type:

• Ratios with a Constant Part — one quantity is untouched while the other changes, so make that part's number equal in both ratios.
• Ratios with a Constant Total — things move between two groups but nothing enters or leaves, so the totals must match.
• Ratios with a Constant Difference — both grow by the same amount, as ages do, so the gap between them is the invariant.

Before-and-After Ratio Problems puts them together. Two people share money 5 : 3. After one gives 12 away, they are equal. The difference of 2 parts was 12 before, and the whole 12 moved from the larger share to nobody, so one part is 6 and the shares were 30 and 18.

Scaling area and volume

Scaling Area is the result people are surprised by twice. Scaling every length by 3 multiplies area by $3^2=9$3² = 9, because both the length and the width tripled. Volume goes by the cube.

This is why a map with a scale of 1 : 50,000 has an area scale of 1 : 2,500,000,000, and why doubling the width of a cake tin needs four times the batter rather than two.

How do you tell direct from inverse in a word problem?

Ask what happens to the second quantity when the first one doubles. If it doubles too, the relationship is direct; if it halves, the relationship is inverse. That single test is more reliable than looking for keywords, because the same words appear in both kinds of question.

Then check the invariant. In a direct relationship every pair of values gives the same quotient; in an inverse one, the same product. If neither is constant across the data you were given, the relationship is not proportional at all, and no amount of scaling will make it behave.

The four mistakes worth naming

• Dividing by one part instead of the total. Sharing in 4 : 5 means dividing by 9. Write the total down first, every time.
• Averaging two speeds. Total distance over total time. The slow leg lasts longer, so it counts for more.
• Converting one half of a compound unit. km/h to m/s needs both the top and the bottom.
• Scaling area by the length factor. Lengths go by k, areas by k², volumes by k³.

Where this leads next

Ratio is fractions used to compare parts rather than describe them, so equivalent fractions and mixed numbers is the topic underneath it, and percentages is ratio with the total fixed at a hundred. Naming the constant of proportionality is the first step into algebraic notation, and the straight line through the origin is where equations take over.

Practise it in the game

Math Challenge teaches each rung above as an illustrated lesson — the bar model drawn out, the before-and-after invariant highlighted, the compound-unit conversion done in two visible steps — inside a catalog of 800+ lessons.

Practise ratio free →