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SSC CGL Maths · Ratio, Proportion and Variation

Ratio Basics

A ratio a : b says the two quantities are a parts and b parts of one common unit, so every ratio question comes down to finding that unit once.

All Ratio, Proportion and Variation notes

Why this matters

Thirty-four questions in the bank, thirty-one of them EASY: twelve turn fractions, percentages or equal parts into a ratio, eleven share a quantity, four count coins and seven find an expression's value from a ratio. Every later page in this chapter uses the ratio unit taught here.

Concept 1 of 4: Turning fractions, percentages and equal parts into a ratio

A ratio does not change when both terms are multiplied or divided by the same number, just as a recipe for 4 tastes the same as the same recipe doubled for 8. So any messy ratio can be scaled until its terms are small whole numbers.

Definition

  • Lowest terms: divide every term by their highest common factor. 18 : 24 : 30 is 3 : 4 : 5.
  • Fractions: multiply every term by the LCM of the denominators. 12:23\frac{1}{2} : \frac{2}{3} times 6 is 3 : 4.
  • Decimals: multiply by 10 or 100 until the decimals go. 0.6 : 1.5 is 6 : 15 = 2 : 5.
  • Comparing two ratios: write each as a fraction and cross-multiply, or bring them to a common denominator.
  • Each term changes by its own percentage: multiply each term by its own factor (a 15% rise is ×1.15\times 1.15), then clear the decimals.
  • Equal parts: if pA=qB=rCpA = qB = rC, then A:B:C=1p:1q:1rA : B : C = \frac{1}{p} : \frac{1}{q} : \frac{1}{r}. For two quantities this is the fast route A:B=q:pA : B = q : p: the bigger multiplier sits with the smaller quantity.

Scaling and equal parts

a:b=ka:kbpA=qB⇒A:B=q:ppA=qB=rC⇒A:B:C=1p:1q:1ra : b = ka : kb \qquad pA = qB \Rightarrow A : B = q : p \qquad pA = qB = rC \Rightarrow A : B : C = \frac{1}{p} : \frac{1}{q} : \frac{1}{r}
  • kany non-zero number
  • p, q, rthe multipliers in the equal statement

Worked example

If 30% of P = 0.75 of Q = 25\frac{2}{5} of R, find P : Q : R.
Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 12 Sep 2025, Shift 3 · Q51Easy

Example 1 · Ratio, Proportion and Variation · Ratio Basics

If 60%60\% of A = 0.3 of B = 16\frac{1}{6} of C, find A: B: C.

Putting the multipliers straight into the ratio

From 2A = 3B the ratio A : B is 3 : 2, not 2 : 3. A quantity multiplied by a bigger number must itself be smaller to give the same product.

Adding the percentage to the ratio term

Terms 6 and 4 raised by 20% and 40% become 6×1.2=7.26 \times 1.2 = 7.2 and 4×1.4=5.64 \times 1.4 = 5.6. Writing 26 : 44 or 6.2 : 4.4 adds the percentage instead of multiplying by it.
Drill 11 more on turning fractions, percentages and equal parts into a ratio

Concept 2 of 4: Sharing a quantity in a ratio with the ratio unit

In a ratio 3 : 5 : 7, the three amounts are 3, 5 and 7 copies of one hidden amount, the ratio unit. Any single fact about the amounts (their total, one share, or a gap between two shares) tells you that unit, and then every share follows.

Definition

  • Write the amounts as ak,bk,ckak, bk, ck, where kk is one part.
  • Total given: k=totala+b+ck = \frac{\text{total}}{a + b + c}.
  • One share given: k=that shareits termk = \frac{\text{that share}}{\text{its term}}.
  • A difference given: k=differencedifference of the termsk = \frac{\text{difference}}{\text{difference of the terms}}.
  • Part to whole: boys : all students = 7 : 17 means boys : girls = 7 : 10.
  • Fast route: a share is its fraction of the total, aa+b+c×total\frac{a}{a + b + c} \times \text{total}, and a gap between two shares is c−aa+b+c×total\frac{c - a}{a + b + c} \times \text{total}.

Share and ratio unit

share of A=aa+b+c×totalk=differencec−a\text{share of A} = \frac{a}{a + b + c} \times \text{total} \qquad k = \frac{\text{difference}}{c - a}
  • a, b, cthe terms of the ratio
  • kthe ratio unit, one part

Worked example

₹4,500 is shared among P, Q and R in the ratio 3 : 5 : 7. How much does R get, and how much more does R get than P?
Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 21 Sep 2025, Shift 2 · Q58Easy

Example 2 · Ratio, Proportion and Variation · Ratio Basics

Three friends share a winning prize in the ratio 2:3:4. If the total prize is Rs. 9000, how much more does the person with the highest share get than the lowest?

Dividing a gap by the total parts

If two numbers in the ratio 4 : 9 differ by 35, the 35 is the gap of 5 parts. Dividing it by 13, the total of the terms, gives a wrong unit.

Reading a part-to-whole ratio as part to part

Boys : total = 7 : 17 does not mean boys : girls = 7 : 17. The girls are the other 10 parts of the 17.
Drill 10 more on sharing a quantity in a ratio with the ratio unit

Concept 3 of 4: Coins in a ratio: number of coins against their value

A ratio of coins counts coins, not rupees. Ten ₹5 coins and ten ₹1 coins are equal in number but very unequal in money, so the count ratio must be turned into money before it meets the total.

Definition

  • If the coins number ak,bk,ckak, bk, ck and are worth v1,v2,v3v_1, v_2, v_3 rupees each, their value is k(av1+bv2+cv3)k(av_1 + bv_2 + cv_3).
  • Set that value equal to the total in the bag to find kk, then the number of each coin.
  • Write paise as rupees: 50 p is ₹0.50, 25 p is ₹0.25, 20 p is ₹0.20.
  • The ratio of the VALUES is the count term times the coin value: ₹10 and ₹5 coins in the number ratio 1 : 4 have values in the ratio 10 : 20 = 1 : 2.
  • Option check: the counts must be whole numbers in the given ratio, which removes most options at sight.

Value of coins in a ratio

total value=k(av1+bv2+cv3)\text{total value} = k(av_1 + bv_2 + cv_3)
  • a, b, cthe ratio of the NUMBERS of coins
  • v_1, v_2, v_3the value of one coin of each kind, in rupees

Worked example

A purse has ₹1, ₹2 and ₹5 coins in the ratio 4 : 3 : 2. The total is ₹240. How many ₹2 coins are there?
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 · Q52Easy

Example 3 · Ratio, Proportion and Variation · Ratio Basics

A bag contains ₹10, ₹5, and ₹2 coins in the ratio 1 : 2 : 3. If the total amount of money in the bag is ₹390, find the number of coins of each kind.

Sharing the money in the coin ratio

If coins are in the number ratio 2 : 3 : 5, the rupees are not split 2 : 3 : 5. Multiply each count term by its coin value first; only then does the total fix the unit.
Drill 3 more on coins in a ratio: number of coins against their value

Concept 4 of 4: Value of an expression when a ratio is given

If a : b is fixed, then a and b are the same multiple of two known numbers. In an expression where every term has the same power, that multiple cancels from top and bottom, so you may simply put the ratio terms in.

Definition

  • Substitute: if a : b = m : n, put a = m and b = n in any expression whose terms all have the same power, such as 6a+2b6a−2b\frac{6a + 2b}{6a - 2b} or a3+b3a3−b3\frac{a^3 + b^3}{a^3 - b^3}.
  • Safe only when the powers match. a2+ba\frac{a^2 + b}{a} is NOT fixed by a : b, because the multiple does not cancel.
  • Working backwards: cross-multiply the given equation and collect terms. From x−2yx+2y=13\frac{x - 2y}{x + 2y} = \frac{1}{3}: 3x−6y=x+2y3x - 6y = x + 2y, so x : y = 4 : 1.
  • Componendo and dividendo: if x+yx−y=pq\frac{x + y}{x - y} = \frac{p}{q}, then xy=p+qp−q\frac{x}{y} = \frac{p + q}{p - q}.
  • Equal ratios add: if a : b = c : d = e : f = m : n, then (pa + qc + re) : (pb + qd + rf) = m : n for any p, q, r.

Rules for given ratios

x+yx−y=pq⇒xy=p+qp−qab=cd=k⇒pa+qcpb+qd=k\frac{x + y}{x - y} = \frac{p}{q} \Rightarrow \frac{x}{y} = \frac{p + q}{p - q} \qquad \frac{a}{b} = \frac{c}{d} = k \Rightarrow \frac{pa + qc}{pb + qd} = k

Worked example

If m : n = 3 : 2, find (4m−n):(2m+3n)(4m - n) : (2m + 3n).
Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 19 Sep 2025, Shift 3 · Q51Moderate

Example 4 · Ratio, Proportion and Variation · Ratio Basics

If (x+y):(x−y)=5:2(x+y):(x-y) = 5:2, find (x3+y3):(x3−y3)(x^3 + y^3) : (x^3 - y^3)

Substituting when the powers differ

a : b = 1 : 2 fixes a+ba−b\frac{a + b}{a - b}, but not a2+ba\frac{a^2 + b}{a}: with a = 1, b = 2 it is 3, and with a = 2, b = 4 it is 4. Substitution is safe only when every term has the same power.

Stopping at the given ratio

From (x+y):(x−y)=7:3(x + y) : (x - y) = 7 : 3, the ratio x : y is (7+3):(7−3)=5:2(7 + 3) : (7 - 3) = 5 : 2. The given ratio is about x+yx + y and x−yx - y, not about x and y.
Drill 6 more on value of an expression when a ratio is given

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)

  • Turning fractions, percentages and equal parts into a ratio

    Scaling and equal parts

    a:b=ka:kbpA=qB⇒A:B=q:ppA=qB=rC⇒A:B:C=1p:1q:1ra : b = ka : kb \qquad pA = qB \Rightarrow A : B = q : p \qquad pA = qB = rC \Rightarrow A : B : C = \frac{1}{p} : \frac{1}{q} : \frac{1}{r}
  • Sharing a quantity in a ratio with the ratio unit

    Share and ratio unit

    share of A=aa+b+c×totalk=differencec−a\text{share of A} = \frac{a}{a + b + c} \times \text{total} \qquad k = \frac{\text{difference}}{c - a}
  • Coins in a ratio: number of coins against their value

    Value of coins in a ratio

    total value=k(av1+bv2+cv3)\text{total value} = k(av_1 + bv_2 + cv_3)
  • Value of an expression when a ratio is given

    Rules for given ratios

    x+yx−y=pq⇒xy=p+qp−qab=cd=k⇒pa+qcpb+qd=k\frac{x + y}{x - y} = \frac{p}{q} \Rightarrow \frac{x}{y} = \frac{p + q}{p - q} \qquad \frac{a}{b} = \frac{c}{d} = k \Rightarrow \frac{pa + qc}{pb + qd} = k

Watch out for (7)

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.

Take the Ratio, Proportion and Variation chapter testDrill all 34 questions