Ratios and mixtures

A ratio is a recipe of parts

A ratio of $3 : 5$ says nothing about size; it says the quantities are $3k$ and $5k$ for some positive number $k$. Introducing that one multiplier turns every ratio question into ordinary algebra. The total is $8k$, so the first quantity is $\frac{3}{8}$ of the whole and the second $\frac{5}{8}$. A ratio alone never fixes the actual numbers; you need one real quantity (a total, a difference, one of the parts) to find $k$.

Part-to-part and part-to-whole

"Boys to girls is $3 : 5$" is part-to-part. "Three eighths of the class are boys" is part-to-whole. They carry the same information, but mixing them up is a classic slip: with a ratio of $3 : 5$, boys are $\frac{3}{8}$ of the class, not $\frac{3}{5}$. When a question gives a total, convert to part-to-whole; when it gives a difference, use the parts: the difference between the groups is $5k - 3k = 2k$.

Linking ratios through a common term

If $a : b = 3 : 4$ and $b : c = 6 : 5$, the two ratios describe $b$ with different numbers of parts. Scale each so that $b$ gets the same number, the LCM of 4 and 6, which is 12: $a : b = 9 : 12$ and $b : c = 12 : 10$. Now $a : b : c = 9 : 12 : 10$. Joining the ratios without this step, as $3 : 4 : 5$, is wrong because the "4" and the "6" are not the same amount of $b$.

When the quantities change

Questions often add or remove items and give the new ratio. Write the original quantities with $k$, apply the change, set the new ratio and cross-multiply. Whatever does not change is your anchor. If men and women are $5 : 3$ and only women join, the number of men is fixed, so the new ratio tells you how many women there must be per man.

Mixtures: track the pure ingredient

A concentration is a part-to-whole ratio: $\frac{\text{amount of ingredient}}{\text{total amount}}$. Before writing any equation, ask which amount stays fixed.

• Adding water (or any other diluting liquid) leaves the amount of the ingredient unchanged. 40 litres of a 20% salt solution hold 8 litres of salt; to make it 16%, the total must become $\frac{8}{0.16} = 50$ litres, so add 10 litres of water.
• Adding pure ingredient raises both the ingredient and the total, so both appear in the equation.
• Removing water (evaporation) keeps the ingredient and lowers the total.

Weighted averages and the balance method

When two groups with different values are combined, the result is a weighted average: it lies between the two values, closer to the value of the larger group. Picture a seesaw. With low value $L$, high value $H$ and mixed value $M$,

$$\frac{\text{amount at } L}{\text{amount at } H} = \frac{H - M}{M - L}$$

The amounts are in the inverse ratio of the distances. Mixing 30% and 75% solutions to get 45%: the distances are $45 - 30 = 15$ and $75 - 45 = 30$, so the amounts of 30% and 75% solution are in the ratio $30 : 15 = 2 : 1$. The 30% solution is the one the mixture sits closer to, so there must be more of it. The same method handles prices of blended goods, average scores of two classes and average speeds over equal times.

Two checks before you choose

Is the mixed value between the two ingredient values, and is it closer to the one there is more of? Does your answer give the stated total? Both catch reversed ratios, the most common mistake in this topic.