Proportion and Ratio Problems

Direct Proportion #

Double one and the other doubles.

In direct proportion doubling one quantity doubles the other

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1 costs 3, 2 cost 6, 3 cost 9. Each step adds the same 3.

Double the count and the cost doubles too. That is direct proportion.

Inverse Proportion #

Double one and the other halves.

In inverse proportion doubling one quantity halves the other

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2 workers take 12 days. 4 workers take only 6.

More workers, fewer days — but 2 × 12 = 4 × 6 = 6 × 4 = 24 every time.

The Constant of Proportionality #

The fixed number the table hides.

Proportional quantities hide one fixed number, and k is its name

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Each y is its x times 3. One fixed number hides in the whole table.

Name it: y = kx, and here k = 3 — the constant of proportionality.

The symbol $\propto$∝ makes the claim without naming k: $y\propto x$y ∝ x means y = kx for some k.

Inverse proportion hides a constant too — the product: x × y = 24 every time.

Proportion by Unit Value #

Find one, then scale to however many you need.

Work out what one is worth and then scale up to however many you need

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4 cost 20, so 20 ÷ 4 = 5 for one of them.

Now scale up: 7 × 5 = 35, so 7 of them cost 35.

Rewriting a Ratio to Match a Share #

Finer parts, the very same amounts.

Cutting every part finer renames a ratio without moving anything

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Ann has 2 parts and Ben has 1: 2 + 1 = 3 parts between them.

Cut every part in two. The bar is the same length — only the counting changed.

So 2 : 1 and 4 : 2 are one ratio. Rewriting it is how you make a share match.

Ratios with a Constant Part #

Anchor to the share that never moves.

If one share never changes you can anchor the whole problem to it

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Ann reads 4 parts and Ben 3. Ann never moved: she is still the 12 she started with.

Ann is the anchor. Her 12 spread over her 4 parts is what prices one part.

Every other share is then just a count of those parts. Ben holds 3, so Ben is 9.

Ratios with a Constant Total #

Nothing in or out, so the total anchors it.

If nothing enters or leaves then the total is your anchor

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Split 3 : 5, some move across — the bars end level: still eight parts in total.

Ratios with a Constant Difference #

Both grow equally, so the gap holds.

If both shares grow by the same amount then the gap between them is fixed

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3 and 7 both gain 2, making 5 and 9 — and both gap brackets still measure 4.

Before-and-After Ratio Problems #

Spot the quantity that did not move.

A ratio problem gets easy once you spot which quantity did not move

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We are told the second share did not change: same 5 parts, same size of part.

So the first share went 2 parts to 4: a gain of 4 − 2 = 2 parts.

Map Scales #

What one map unit stands for on the ground.

A scale tells you how much real distance one map unit stands for

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At 1 cm to 10 km, those 5 map centimeters are 5 × 10 = 50 km on the ground.

Scaling Area #

Double the sides and the area goes up four times.

Doubling every side multiplies the area by four rather than by two

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2 × 3 = 6 squares fit inside this rectangle.

Double both sides and 4 × 6 = 24 squares fit — 4 times as many.

Each direction picks up its own k, so area grows k × k: doubling 4, tripling 9.