SSC CGL Maths · Ratio, Proportion and Variation

Mixture Quantities and Replacement

A mixture question is bookkeeping: write how much of each ingredient the vessel holds, change only what the stem changes, and form the new ratio or percentage at the end.

All Ratio, Proportion and Variation notes

Why this matters

Thirty-three questions in the bank, fourteen MODERATE and the chapter's only HARD one. Fifteen add a single ingredient, five combine two mixtures, ten take out part of a mixture before topping it up, and three repeat a replacement. The fast routes here come from spotting the ingredient that does not change.

Concept 1 of 4: Adding one ingredient while the other stays fixed

Pouring water into milk changes the water and the total, but not a drop of the milk. So the milk is a fixed anchor: its amount and the target ratio or percentage tell you exactly what the water and the total must become.

Definition

A mixture of total T with A : B = a : b holds $A = \frac{a}{a + b} \times T$ and B is the rest.
Adding only B leaves A fixed. From A and the target, find the new B (or the new total) and subtract.
Percentage form: if 12 litres of water must become 15% of the solution, the new total is $\frac{12}{0.15} = 80$ litres.
Adding the ingredient whose share you want to raise: $\frac{A + x}{T + x} = p$ , where p is the target fraction.
Fast route with ratio units: scale the old and new ratios so the fixed ingredient has the same term in both. The rise in the other term, in units, is the amount added.
Adding to both ingredients at once: write the equation $\frac{A + x}{B + y} = \frac{m}{n}$ .

Adding to a mixture

\text{new total} = \frac{\text{fixed amount}}{\text{its new fraction}} \qquad \frac{A + x}{T + x} = p

A, Tthe amount of the ingredient being added to, and the total
pthe target fraction of that ingredient

Worked example

A 48-litre mixture has milk and water in the ratio 5 : 3. How much water must be added to make the ratio 5 : 4?

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 17 Sep 2025, Shift 2 · Q60Easy

Example 1 · Ratio, Proportion and Variation · Mixture Quantities and Replacement

A barrel contains a mixture of alcohol and water in the ratio 7:4 respectively. If 12 litres of water are added to it, the ratio of alcohol to water becomes 7:6. Find the initial total quantity of the mixture in the barrel.

Working from the total instead of the fixed ingredient

When only water is added, the total and the water both change, but the milk does not. Start from the milk; a percentage of the old total is the wrong anchor.

Adding litres to a ratio term

Adding 10 litres of water to a 5 : 3 mixture does not make it 5 : 13. Turn the ratio into litres first, add the 10 litres, and then form the new ratio.

Drill 14 more on adding one ingredient while the other stays fixed

Concept 2 of 4: Combining two mixtures

When two jugs are poured into one, the milk from both ends up together and the water from both ends up together. So add the milk to the milk and the water to the water, and only then compare them.

Definition

From each mixture, an ingredient's amount is its fraction times the amount taken: from 8 litres of a 3 : 1 mix, milk is $\frac{3}{4} \times 8 = 6$ litres.
Add each ingredient across the mixtures, then form the new ratio.
Equal volumes: the new fraction of an ingredient is the plain average of its fractions in the two. Unequal volumes need the amounts.
The same solution in two containers has the same ratio in both, whatever the amounts.
Comparing two mixtures: compare the fractions, not the amounts.

Combined amount of an ingredient

A_{\text{new}} = \frac{a_1}{a_1 + b_1}V_1 + \frac{a_2}{a_2 + b_2}V_2

a : bthe ratio in each mixture
V_1, V_2the amounts taken from each

Worked example

Vessel P holds milk and water in the ratio 3 : 1 and vessel Q holds them in the ratio 2 : 3. 8 litres from P and 10 litres from Q are mixed. What is the ratio of milk to water?

Practice this conceptself-check · 3 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 16 Sep 2025, Shift 1 · Q63Easy

Example 2 · Ratio, Proportion and Variation · Mixture Quantities and Replacement

Two containers, A and B, contain mixtures of alcohol and water. Container A has alcohol and water in the ratio 4:1, while Container B has them in the ratio 5:2. If 10 liters are drawn from Container A and 14 liters from Container B, and the contents are mixed in a third container, what is the ratio of alcohol to water in the new mixture?

Adding the ratio terms of unequal amounts

10 litres of a 1 : 1 mixture and 30 litres of a 3 : 1 mixture do not give 4 : 2. In litres, milk is

5 + 22.5 = 27.5

and water

5 + 7.5 = 12.5

, a ratio of 11 : 5.

Drill 4 more on combining two mixtures

Concept 3 of 4: Taking out part of a mixture and topping it up

A well-stirred mixture is the same all through, so any cupful taken out has the mixture's own ratio. Removing a fifth of the mixture removes a fifth of every ingredient, and then whatever is poured back is simply added.

Definition

Removing x litres from T litres removes each ingredient in the mixture's ratio: each one falls to $(1 - \frac{x}{T})$ of its amount.
Then add what is poured back: water, the pure ingredient, or another mixture (split it into its ingredients first).
Topped up with water: the other ingredient left $=$ its original amount $\times (1 - \frac{x}{T})$ , and the total is T again.
Topped up with the pure ingredient: that ingredient changes twice, so work from the OTHER one, which only falls.
Three ingredients and a stated difference: after the removal, the mixture left is still in the old ratio. Write each ingredient from it, add what is poured in, and use the difference.

Amount left after a removal

\text{left} = \text{original} \times \left(1 - \frac{x}{T}\right)

xthe amount of mixture taken out
Tthe total before it is taken out

Worked example

A 50-litre can holds milk and water in the ratio 4 : 1. 10 litres are taken out and replaced with water. What is the new ratio of milk to water?

Practice this conceptself-check · 3 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 21 Sep 2025, Shift 2 · Q65Moderate

Example 3 · Ratio, Proportion and Variation · Mixture Quantities and Replacement

A mixture of 150 litres contains acid and water in the ratio 2:3. If 30 litres of the mixture is replaced with water, what will be the percentage of acid in the final mixture?

Taking out only one ingredient

Drawing 10 litres from a 4 : 1 milk and water mixture takes out 8 litres of milk and 2 of water, not 10 litres of milk. A removal always carries the mixture's own ratio.

Tracking the ingredient that is poured back

When pure acid is poured back, the acid falls and then rises, which needs two steps. The water only falls, so follow the water and the answer comes in one equation.

Drill 9 more on taking out part of a mixture and topping it up

Concept 4 of 4: Repeated replacement

Each round of draw-and-refill keeps the same fraction of the original liquid, because each draw takes the same share of whatever is there. So the liquid left shrinks by the same factor every round, like a percentage applied again and again.

Definition

If x litres are drawn from T litres and replaced with water, n times, the original liquid left is $A\left(1 - \frac{x}{T}\right)^n$ .
As a percentage: each round keeps $1 - \frac{x}{T}$ of the liquid, so a 60% strength after two rounds of keeping $\frac{4}{5}$ is $60 \times \frac{16}{25} = 38.4\%$ .
Safe only when the same amount is drawn every round, the refill is the same liquid every round, and the total returns to T. If the refill changes between rounds (water, then acid), work round by round.

Repeated replacement

A_n = A\left(1 - \frac{x}{T}\right)^n

Athe amount of the original liquid at the start
nthe number of rounds

Worked example

A 64-litre can is full of milk. 16 litres are taken out and replaced with water, three times in all. How much milk is left?

Practice this conceptself-check · 2 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 15 Sep 2025, Shift 1 · Q63Easy

Example 4 · Ratio, Proportion and Variation · Mixture Quantities and Replacement

A cistern contains 100 liters of pure syrup. 10 liters of syrup are drawn out and replaced by water. Then, 10 liters of the mixture are drawn out and replaced by water again. What is the ratio of syrup to water in the final mixture?

Subtracting the same amount every round

From 40 litres of juice, two draws of 4 litres leave 32.4 litres, not 32. The second draw takes out less juice than the first, because the jar is already weaker.

Drill 2 more on repeated replacement

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (4)

Adding one ingredient while the other stays fixed
Adding to a mixture
$\text{new total} = \frac{\text{fixed amount}}{\text{its new fraction}} \qquad \frac{A + x}{T + x} = p$
Combining two mixtures
Combined amount of an ingredient
$A_{\text{new}} = \frac{a_1}{a_1 + b_1}V_1 + \frac{a_2}{a_2 + b_2}V_2$
Taking out part of a mixture and topping it up
Amount left after a removal
$\text{left} = \text{original} \times \left(1 - \frac{x}{T}\right)$
Repeated replacement
Repeated replacement
$A_n = A\left(1 - \frac{x}{T}\right)^n$

Watch out for (6)

Working from the total instead of the fixed ingredient
→ Adding one ingredient while the other stays fixed
Adding litres to a ratio term
→ Adding one ingredient while the other stays fixed
Adding the ratio terms of unequal amounts
→ Combining two mixtures
Taking out only one ingredient
→ Taking out part of a mixture and topping it up
Tracking the ingredient that is poured back
→ Taking out part of a mixture and topping it up
Subtracting the same amount every round
→ Repeated replacement

Test yourself on Ratio, Proportion and Variation

20 past SSC CGL questions from this chapter, timed at 12 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

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