Ratio Based Age Problems: Formulas, Methods and Examples

Ratio Based Age Problems use the relationship between people’s ages to find present, past or future ages. The main method is to represent ages as multiples of the ratio terms, then apply the given age difference, sum or time condition. This page covers age ratio formulas, equation methods, ratio age tricks and solved age word problems.

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What Are Ratio Based Age Problems?

Ratio Based Age Problems are questions in which two or more ages are expressed in a ratio, such as 3:5. If the common multiplier is x, the corresponding ages are 3x and 5x.

For a ratio a:b, the ages can be written as ax and bx. If their age difference is d, then x(a-b)=d, so x=d/(a-b). If their age sum is s, then x(a+b)=s, so x=s/(a+b). The same difference remains constant in the past and future, while both ages change by the same number of years.

Ratio Based Age Problems Formula & Tricks

Important Formulas

Ages from a ratio
Ages = ax and bx for ratio a:b

Use a common multiplier x for the two ratio terms.

Using age difference
x = d/(a-b)

If the age difference is d, divide it by the difference between the ratio terms.

Using age sum
x = s/(a+b)

If the sum of the ages is s, divide it by the sum of the ratio terms.

Future age ratio
(ax+t)/(bx+t) = p/q

After t years, add t to both present ages before applying the new ratio.

Past age ratio
(ax-t)/(bx-t) = p/q

t years ago, subtract t from both present ages before applying the old ratio.

Quick Tricks

Use the ratio difference directly

When the age difference is given, one ratio part equals the actual difference divided by the difference between the ratio terms.

Example: If the ratio is 4:7 and the age difference is 15 years, one part is 15/(7-4)=5. The ages are 20 and 35 years.
Keep the age difference unchanged

Adding or subtracting the same number of years from two ages does not change their difference.

Example: If two ages are 24 and 36, their difference is 12 now and also 12 years later: 36 and 48.
Check a proposed ratio by cross multiplication

For ages A and B in the ratio p:q, verify the result using qA=pB or A/B=p/q.

Example: Ages 28 and 42 have ratio 28:42=2:3 because 3×28=2×42.

Ratio Based Age Problems Concepts

Finding Ages from Present Age Ratio and Difference

Write the ages as ax and bx, then determine x from the given difference.

For ratio a:b and age difference d, |b-a|x=d. Therefore, x=d/|b-a|. Multiply x by each ratio term to obtain the ages. The larger ratio term represents the older person when the ratio is stated in the same order as the ages.

Example: The ages of A and B are in the ratio 5:8, and their difference is 12 years. Here, 3x=12, so x=4. Their ages are 20 and 32 years.

Finding Ages from Present Age Ratio and Sum

When the sum of two ages is known, divide the sum by the sum of the ratio terms.

If the ages are ax and bx and their total is s, then (a+b)x=s. Hence, x=s/(a+b), and the ages are ax and bx. The calculated ages should be checked by adding them.

Example: Two ages are in the ratio 3:5 and their sum is 64. Thus, 8x=64, x=8, and the ages are 24 and 40 years.

Past and Future Age Ratio Equations

For a time change of t years, add t to both ages for the future and subtract t from both ages for the past.

If present ages are ax and bx, a future ratio p:q gives (ax+t)/(bx+t)=p/q. A past ratio p:q gives (ax-t)/(bx-t)=p/q. Cross multiplication converts either relation into a linear equation in x.

Example: Five years from now, A:B will be 4:5, and A is currently 5 years younger than B. Let present ages be A and A+5. Then (A+5):(A+10)=4:5. Therefore, 5A+50=4A+20, so A=-30, which is impossible; hence these conditions cannot describe valid present ages.

Comparing Ratios at Different Times

Ages do not retain the same ratio after equal years are added or subtracted unless the two ages are equal.

If present ages are ax and bx, their future ratio is (ax+t):(bx+t), not a:b. To solve a question with two time conditions, express the ages using one variable and form an equation from the stated ratio. Always ensure every past age remains positive.

Example: A father and son are currently in the ratio 5:2. If their age difference is 27 years, one part is 9, so their present ages are 45 and 18. After 9 years, their ratio becomes 54:27=2:1, not 5:2.

Ratio Based Age Problems Video Lessons

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Age Problems: Ratios and Two-Point Questions

Learn to solve age problems using ratios and information about two points in time, including relationships between present, past, and future ages.

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Practice Ratio Based Age Problems Questions

Practise published questions related to this topic.

1The HCF and LCM of two positive integers are 36 and 7,560, respectively. If one of the numbers is 540, what is the other number?→ 2Selling price of an article is Rs. 4000. If the value of the loss is Rs. 1000, then what is the loss percentage?→ 3If x = √(1671 + √(93 + √49)) , then what is the value of x?→ 4Two numbers are in the ratio of 4 : 5, and their HCF is 16. Find the LCM of the two numbers.→ 5Simplify: (10 × a⁵ × b⁶)/(2 × a² × b³)→ 6If a four-digit number 34X1 is divisible by 7, then X2 is:→ 7Find the least number which leaves remainder 3 when divided by 5, 7, and 8.→ 8The price of a laptop was initially $1,200. During a sale, the price was first reduced by 20%, and then an additional discount of 15% was applied to the new price. What is the final price of the laptop?→ 9A person sells an article at the profit of 60 percent. Cost price and selling price are increased by 30 percent and 'y' percent respectively. If the new profit percentage is 40 percent, then what is the value of 'y'?→ 10Simplify: √(√(24² - 16² + 2²) + √(2² × 3² + √169))→

Ratio Based Age Problems Quick Quiz

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Quick Revision Notes

Ratio Based Age Problems: Quick Revision

Use these formulas and rules to solve age ratio questions.

  • For ratio a:b, write the ages as ax and bx.
  • With difference d, x=d/|a-b|.
  • With sum s, x=s/(a+b).
  • Add t years to both ages for a future condition.
  • Subtract t years from both ages for a past condition.
  • The difference between two ages remains constant over time.
  • Use cross multiplication to solve equations involving age ratios.
  • Check that all calculated ages are positive and satisfy the original condition.

Ratio Based Age Problems FAQs

What is the basic age ratio formula?

If two ages are in the ratio a:b, write them as ax and bx, where x is the common multiplier.

How do you find ages when the ratio and difference are given?

For ratio a:b and difference d, calculate x=d/|a-b|. The ages are ax and bx. For example, ratio 2:5 and difference 18 gives x=6 and ages 12 and 30.

How do you find ages when the ratio and sum are given?

For ratio a:b and sum s, use x=s/(a+b). For a ratio 3:7 and sum 50, x=5, so the ages are 15 and 35.

How are future age ratio questions solved?

If present ages are ax and bx, use (ax+t)/(bx+t)=p/q for a ratio p:q after t years. Cross multiply to find x.

How are past age ratio questions solved?

If present ages are ax and bx, use (ax-t)/(bx-t)=p/q for a ratio p:q t years ago. Both ages must remain positive after subtraction.

Does the difference between two ages change with time?

No. If the present ages are A and B, their difference remains |A-B| after the same number of years is added to or subtracted from both.

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