SSC CGL Reasoning · Arithmetic Reasoning

Ratio and Mixtures

A ratio splits a total into equal parts; find the size of one part, and in a mixture keep fixed the ingredient that is not being added.

All Arithmetic Reasoning notes

Why this matters

Ten questions in the bank, eight EASY and two MODERATE, all from the 2025 shifts. Three split a total in a ratio, two chain two ratios or share a profit, and five add water or milk to a mixture to reach a new ratio.

Concept 1 of 3: Splitting a total in a ratio

A ratio of 3 : 5 means the total is cut into 8 equal parts, 3 for one and 5 for the other. Once you know the size of one part, every number in the question follows.

Definition

Ratio a : b with total T: one part $= T \div (a + b)$ . The numbers are $a$ parts and $b$ parts.
Difference given instead: one part $=$ difference $\div (b - a)$ .
One number given: one part $=$ that number divided by its own term of the ratio.
Option elimination: the right pair must add up to the total AND be in the ratio. SSC builds wrong options that add up correctly, so check the ratio too. A quick ratio check: both numbers must be multiples of the same part.

Size of one part

\text{One part} = \frac{T}{a + b} \qquad \text{One part} = \frac{\text{Difference}}{b - a}

a : bthe ratio
Tthe total

Worked example

Two numbers are in the ratio 5 : 8 and their sum is 195. Find the numbers.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 12 Sep 2025, Shift 2 · Q23Easy

Example 1 · Arithmetic Reasoning · Ratio and Mixtures

Two numbers are in ratio 7:9, and their sum is 128. Find the numbers.

Dividing the total by one term

For a ratio of 3 : 5 and a total of 96, one part is

96 \div 8 = 12

, not

96 \div 3

96 \div 5

. Divide by the sum of the terms.

Checking only the sum

Wrong options are often built to add up to the total. A pair that adds up correctly but is not in the ratio is still wrong.

Drill 2 more on splitting a total in a ratio

Concept 2 of 3: Chained ratios and profit shares

Two ratios that share a middle term can be joined into one, but only after the middle term is made the same in both. Partners share profit in the ratio of what each put in and for how long.

Definition

A : B and B : C: multiply each ratio so that B is the same number in both, then read A : B : C. Or multiply fractions: $\frac{A}{C} = \frac{A}{B} \times \frac{B}{C}$ .
Partnership: profit is shared in the ratio of capital $\times$ time for each partner.
If every partner invests for the same time, the ratio of capitals alone decides the shares.
Once the ratio is known, one partner's share gives the size of one part, as in any ratio split.

Joining ratios and sharing profit

\frac{A}{C} = \frac{A}{B} \times \frac{B}{C} \qquad \text{Share} \propto \text{Capital} \times \text{Time}

Worked example

If A : B = 3 : 4 and B : C = 6 : 7, find A : B : C.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 19 Sep 2025, Shift 1 · Q19Easy

Example 2 · Arithmetic Reasoning · Ratio and Mixtures

A and B are partners in a business. A invests ₹20,000 for a period of 2 years. B invests ₹30,000 for a period of 3 years. If A's share of the total profit is ₹4,000, what is B's share of the profit?

Joining ratios without matching the middle term

A : B = 2 : 3 and B : C = 4 : 5 do not give A : C = 2 : 5. Make B equal first (12 in both): 8 : 12 : 15, so A : C = 8 : 15.

Ignoring time in a partnership

When partners invest for different lengths of time, share the profit by capital times time. The capitals alone decide only when the times are equal.

Drill 1 more on chained ratios and profit shares

Concept 3 of 3: Adding to a mixture

When you pour water into a mixture, the other ingredient does not change. Hold that fixed amount steady, scale the new ratio to it, and the amount to add is the gap.

Definition

Step 1: split the mixture into its two parts using the ratio.
Step 2: the part you do not add stays the same. Use the new ratio to find what the added part must become.
Step 3: amount to add $=$ new amount $-$ old amount of that part.
Fast route in units: work in ratio units. Keep the unchanged part's units equal in the old and new ratio, read how many units the added part gains, and multiply by the size of one unit (total $\div$ units in the old ratio).
This works when one part is added and the other is left alone. If some mixture is taken out and replaced, both parts change and you need a different method.

Amount to add

\text{New amount of added part} = \text{Fixed part} \times \frac{\text{its new ratio term}}{\text{fixed part's new ratio term}}

Worked example

A 45 L mixture has juice and water in the ratio 7 : 2. How much water must be added to make the ratio 7 : 3?

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 17 Sep 2025, Shift 1 · Q22Moderate

Example 3 · Arithmetic Reasoning · Ratio and Mixtures

A 30 L mixture contains milk and water in the ratio 3:2. How much milk must be added to make it 4:1?

Changing the part that was not added

Adding water leaves the milk exactly as it was. Scale the new ratio to the unchanged part, never to the old total.

Giving the new amount instead of the amount added

If the water must become 15 L and there is already 10 L, the answer is the 5 L poured in, not 15 L.

Drill 4 more on adding to a mixture

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 (3)

Splitting a total in a ratio
Size of one part
$\text{One part} = \frac{T}{a + b} \qquad \text{One part} = \frac{\text{Difference}}{b - a}$
Chained ratios and profit shares
Joining ratios and sharing profit
$\frac{A}{C} = \frac{A}{B} \times \frac{B}{C} \qquad \text{Share} \propto \text{Capital} \times \text{Time}$
Adding to a mixture
Amount to add
$\text{New amount of added part} = \text{Fixed part} \times \frac{\text{its new ratio term}}{\text{fixed part's new ratio term}}$

Watch out for (6)

Dividing the total by one term
→ Splitting a total in a ratio
Checking only the sum
→ Splitting a total in a ratio
Joining ratios without matching the middle term
→ Chained ratios and profit shares
Ignoring time in a partnership
→ Chained ratios and profit shares
Changing the part that was not added
→ Adding to a mixture
Giving the new amount instead of the amount added
→ Adding to a mixture

Test yourself on Arithmetic Reasoning

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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