HCF & LCM · Topic 3 of 6

⚖️ Numbers from a Ratio

2 exam question types, fully solved

When a question gives you the ratio of two or three numbers plus either their HCF or LCM, you can rebuild the actual numbers in one move — no guessing.

Ratio + HCF or LCM ⇒ the actual numbers

If the numbers are in the ratio with co-prime, then a single common multiplier makes them and . That multiplier is the HCF.

• Key relation: . So from a given LCM, ; from a given HCF, just multiply back.

• Three numbers, watch out: . The LCM of the ratio terms equals their product only when they are pairwise co-prime — e.g. , not .

• Fraction ratio? Multiply every term by the LCM of the denominators first to get whole-number ratio terms, then proceed.

⚖️HCF & LCM MachineSame prime factors — lowest powers make the HCF, highest powers make the LCM

First

Second

Third (optional)

72=2³×3²

120=2³×3×5

HCF → only the primes shared by every number (highlighted), each at its LOWEST power. LCM → every prime that appears anywhere, each at its HIGHEST power.

HCF=2³×3=24

LCM=2³×3²×5=360

Check: HCF × LCM = 24 × 360 = 8,640
a × b = 72 × 120 = 8,640 — they match ✓

This identity works for exactly two numbers — it fails for three or more.

Two numbers in ratio with HCF 5 are and .

Q1Two numbers are in ratio 2:3 and their LCM is 48. Their HCF is:

Q2Two numbers are in ratio 5:6 with HCF 4. The larger number is:

Q3Three numbers in ratio 2:3:4 have HCF 5. Their LCM is:

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Real exam questions — Numbers from a Ratio

2 question types · 8 solved examples from real SSC papers

In a ratio a : b (co-prime), the common multiplier is the HCF and the LCM is a·b times it. Reduce the ratio first, then scale.

How to solve this type

If two numbers are in the ratio

a:b

with

a,b

co-prime, their HCF is the common multiplier

x

, so the numbers are

ax

and

bx

. Their LCM is

ab\cdot x

. So: LCM

=ab\times\text{HCF}

. From a given LCM,

x=\dfrac{\text{LCM}}{ab}

; from a given HCF, just multiply back. Always reduce the ratio to co-prime terms first.

Two numbers are in the ratio 3 : 4 and their LCM is 60. Find the numbers.

A9 and 12B12 and 16C15 and 20D6 and 8

Three numbers are in the ratio 3 : 5 : 7 and their LCM is 2625. Find their HCF.

A50B125C100D25

Two numbers are in the ratio 11 : 6 and their HCF is 4. Find the smaller number.

A18B24C30D12

Two numbers are in the ratio 7 : 8 and their LCM is 112. Find their HCF.

A8B2C4D14