EQN-02-05

Mixtures & Combining Ratios

Track a part of a whole through mixing, and chain two ratios into one three-term ratio.

~5 min read•Difficulty 3/5

A part of a mixture

In a mixture given by a ratio, the amount of one ingredient is its share of the total.

A drink is juice : water = 2 : 3. In 30 litres,

total parts = 5, one part = 6 L, so juice = 12 L and water = 18 L .

Changing the mix

If you add only one ingredient, that ingredient's amount changes while the other stays fixed — recompute the ratio from the new amounts, not by adjusting the old ratio directly.

Start 12 L juice + 18 L water . Add 6 L water \to 12 : 24 = 1 : 2.

Chaining ratios (the bridge term)

To merge A : B and B : C, scale them so the shared term B matches, then read all three.

A : B = 2 : 3 B : C = 6 : 7

Scale the first by 2 \to A : B = 4 : 6, matching B = 6

So A : B : C = 4 : 6 : 7

Worked examples

Example 1 (difficulty 2/5)

A 40-litre mixture has acid : water

= 3 : 5.

How many litres of acid does it contain?

◎ Predict the answer first:

Example 2 (difficulty 3/5)

a : b = 2 : 3

and

b : c = 6 : 5,

what is a : c ?

◎ Predict the answer first:

Tips

Convert a mixture ratio to actual amounts (litres, grams) before changing it.
When only one ingredient is added, hold the other fixed and recompute.
To chain ratios, make the shared term equal first.

Common mistakes

Adjusting the old ratio directly after adding one ingredient

Why: It seems faster than recomputing.

Fix: Work in actual amounts, change the one that grew, then form the new ratio.

Multiplying a : b and b : c term-by-term to "combine"

Why: It looks like the natural shortcut.

Fix: Equalise the bridge term b first, then read all three terms.

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