Fundamentals of Business Mathematics and Statistics · Ratios, Variations and Proportions
Applications of Ratios and Proportions: Word Problems
Updated 10 October 2026 · Fact-checked
Applications of ratios and proportions turn a word problem into parts. Write the ratio, assign each quantity a multiple of a common unknown (like 3x and 5x), form one equation from the given total, difference or changed condition, solve for x, then find the required quantity.
Understand Applications of Ratios and Proportions
A ratio compares two quantities of the same kind. If Asha has ₹300 and Ravi has ₹500, their money is in the ratio 3 : 5. The ratio has no unit and only shows how many parts each one has.
The key idea in every word problem is the common multiplier. If two quantities are in the ratio 3 : 5, they are 3x and 5x for some number x. You do not know x yet. The problem always gives one more fact, such as the total, the difference, or one actual value. That fact lets you find x.
A proportion says two ratios are equal, for example a : b = c : d. It is used when a rate stays fixed, such as the cost of 5 kg when you know the cost of 3 kg.
Mixtures, sharing and age problems are the same idea in different clothes. In sharing, the total is split into parts. In mixtures, the quantities of the ingredients are in a ratio. In age problems, the ratio changes after some years, so you add or subtract the same number from each age, never from the ratio terms directly.
Key formulas to remember
- Dividing a sum in a ratio
- Share of first = (a ÷ (a + b)) × Total, for ratio a : b
- For three parts a : b : c, divide by a + b + c.
- Common multiplier method
- Quantities = ax, bx, cx; find x from the given total or difference
- Works for sharing, mixtures and ages.
- Proportion rule
- a : b = c : d ⇔ a × d = b × c
- Product of extremes equals product of means.
- Age problem with change
- (A ± n) : (B ± n) = given ratio
- Add or subtract the same number of years to both actual ages.
- Mixture quantity
- Quantity of an ingredient = (its part ÷ total parts) × total mixture
- Use only when the ratio is by the same measure (all litres or all kg).
- Change in ratio
- (ax ± p) : (bx ± q) = m : n, so n(ax ± p) = m(bx ± q); solve for x
- Here m : n is the given new ratio. Cross-multiply to get a linear equation in x, then solve for x.
How to solve Applications of Ratios and Proportions questions
Use this method for any sharing, mixture, age or changed-ratio question.
- 1Read the question and list what is asked and what is given (total, difference, or a changed condition).
- 2Write the ratio and set the quantities as ax and bx (or ax, bx, cx).
- 3Translate the given fact into one equation in x. For a total, add the parts. For a difference, subtract them.
- 4For changes (years pass, items added or removed), apply the change to the actual quantities, then set up the new ratio.
- 5Cross-multiply the new ratio and solve the linear equation for x.
- 6Substitute x back to find the quantity the question asks for, not just x.
- 7Check that the answer satisfies the original ratio and the given fact, then match with the options.
Quickest way: Total parts shortcut
When to use it: Use when a total is given and the ratio is fixed, with no change in the ratio.
- Add the ratio terms to get total parts.
- Divide the total by total parts to get the value of one part.
- Multiply by each ratio term to get the shares.
- In MCQs, first check which option is divisible by the total parts, or test the options by adding to the total.
- For age problems, plug in the options: add the given years and see which pair gives the stated ratio.
Common mistakes in Applications of Ratios and Proportions
Adding years to the ratio terms instead of to the actual ages.
Ratio numbers look like ages, so students treat 3 : 5 as 3 and 5 years.
Fix: Always write ages as 3x and 5x first, then form (3x + n) : (5x + n).
Taking one part as the total.
Students divide the total by one ratio term instead of by the sum of terms.
Fix: Divide by a + b (or a + b + c) to get the value of one part.
Mixing units in a ratio, such as rupees with paise or metres with centimetres.
The units are hidden in the wording.
Fix: Convert everything to the same unit before writing the ratio.
Stopping after finding x.
Finding x feels like the final step.
Fix: Re-read the question and compute the exact quantity asked, such as the larger share or the age after some years.
Using the past years with the wrong sign.
Words like 'ago' and 'hence' are overlooked.
Fix: Use minus for years ago, plus for years hence, applied to each person.
Worked examples
Example 1
₹1,20,000 is divided among A, B and C in the ratio 2 : 3 : 7. How much does C get?
Show the solution
- Total parts = 2 + 3 + 7 = 12.
- One part = ₹1,20,000 ÷ 12 = ₹10,000.
- C's share = 7 × ₹10,000 = ₹70,000.
Answer: ₹70,000
Example 2
The ages of Meera and Suresh are in the ratio 4 : 5. After 6 years the ratio becomes 5 : 6. Find Suresh's present age.
Show the solution
- Let present ages be 4x and 5x.
- After 6 years: (4x + 6) : (5x + 6) = 5 : 6.
- Cross-multiply: 6(4x + 6) = 5(5x + 6).
- 24x + 36 = 25x + 30.
- x = 6.
- Suresh's age = 5x = 30 years.
- Check: Meera = 24, after 6 years 30 and 36, ratio 30 : 36 = 5 : 6.
Answer: 30 years
Exam tips
- Look at the options first in age problems. Testing two or three options can be faster than forming the equation.
- Check whether the question asks for the present value, the value after a change, or the difference between shares.
- Convert to common units at the start. Option traps often come from unit slips.
- Use the total-parts check: the total must be a multiple of the sum of ratio terms for whole-number shares.
- There is no negative marking, so attempt every question. Aim for about one minute per question on average (50 questions in 1 hour) and skip lengthy ones to return later.
Practice questions from Ratios, Variations and Proportions
- A quantity y varies directly as x and inversely as the square of z. When x = 12 and z = 2, y = 9. Find y when x = 8 and z = 4.
- Three partners A, B and C invest in the ratio 3 : 4 : 5. A invests for 8 months, B for 6 months and C for 4 months. If the total profit is ₹…
- A factory's 12 machines produce 3,600 units in 15 days working at the same rate. How many units will 20 such machines produce in 9 days?
- If 6, 9 and x are in continued proportion, what is the value of x?
- The ratio of two numbers is 5:3. If 4 is added to each, the ratio becomes 3:2. What is the sum of the original numbers?
Applications of Ratios and Proportions in other exams
The same ground in other exams, if you are preparing for more than one or want another angle on it.
Applications of Ratios and Proportions: frequently asked questions
How do I divide a sum in a given ratio?
Add the ratio terms to get total parts, divide the sum by total parts to find one part, and multiply by each term. For 2 : 3 on ₹500, one part is ₹100, so shares are ₹200 and ₹300.
How do I solve age problems using ratio and proportion?
Write the present ages as ax and bx. Add or subtract the years to each age, form the new ratio equation, cross-multiply and solve for x. Then find the age asked.
What is the difference between ratio and proportion?
A ratio compares two quantities. A proportion states that two ratios are equal. Proportion is used to find a missing term when a rate or relation stays constant.
Can I add the same number to both terms of a ratio and keep it the same?
No. Adding the same number to both terms changes the ratio, unless the two terms are equal. Multiplying or dividing both terms by the same non-zero number keeps the ratio unchanged.