K Method in Ratio and Proportion

K Method in Ratio and Proportion represents quantities with a common multiplier. If two quantities are in the ratio a:b, they can be written as ak and bk. The method helps find unknown quantities, combine related ratios, and solve proportion problems by forming equations in k.

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What is the K Method in Ratio and Proportion?

The K Method expresses quantities in a ratio as the product of their ratio terms and a common multiplier k. If x:y = a:b, then x = ak and y = bk.

Here, k represents the value of one ratio unit. For example, if boys:girls = 3:5, write boys = 3k and girls = 5k. If their total is 64, then 3k + 5k = 64, so 8k = 64 and k = 8. Therefore, the numbers are 24 and 40.

K Method in Ratio and Proportion Video Lessons

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K-Method for Ratio Problems

Learn how to apply the K-Method to solve ratio problems efficiently by representing quantities with a common multiplier and deriving the required values.

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K Method in Ratio and Proportion Formula & Tricks

Important Formulas

Basic K Method Formula
If x:y = a:b, then x = ak and y = bk

The same multiplier k is used for both quantities because their ratio is fixed.

Finding k from a Total
k = Total ÷ (a + b)

For quantities ak and bk, their total is (a + b)k.

Finding k from a Difference
k = Difference ÷ |a − b|

The difference between ak and bk is |a − b|k.

Three-Term Ratio
If x:y:z = a:b:c, then x = ak, y = bk, z = ck

One common multiplier is applied to every term of the ratio.

Direct Proportion
y ∝ x, so y = kx

In direct proportion, the ratio y/x remains constant.

Inverse Proportion
y ∝ 1/x, so xy = k

In inverse proportion, the product of the two quantities remains constant.

Quick Tricks

Use the Sum of Ratio Terms

When the total of quantities in the ratio a:b is given, divide the total by a + b to find k directly.

Example: If two numbers are in the ratio 4:7 and their sum is 99, then k = 99 ÷ 11 = 9. The numbers are 36 and 63.
Use the Difference of Ratio Terms

When the difference is given, divide it by the difference between the ratio terms.

Example: If two numbers are in the ratio 5:8 and differ by 21, then k = 21 ÷ (8 − 5) = 7. The numbers are 35 and 56.
Equalise Common Terms Before Combining Ratios

For two ratios with a common quantity, make the common terms equal before writing the combined ratio.

Example: If A:B = 2:3 and B:C = 4:5, make B equal to 12. The ratios become A:B = 8:12 and B:C = 12:15, so A:B:C = 8:12:15.

K Method in Ratio and Proportion Concepts

Finding Quantities from a Given Total

Write the quantities as ak and bk, add them, and solve for k.

If x:y = a:b and x + y = T, then ak + bk = T. Thus, k = T/(a + b), followed by x = ak and y = bk.

Example: The ratio of red balls to blue balls is 3:2, and there are 45 balls in total. Let the numbers be 3k and 2k. Then 5k = 45, so k = 9. Red balls = 27 and blue balls = 18.

Finding Quantities from a Given Difference

Use the difference between the ratio terms to determine k.

If x:y = a:b and x − y = D, then |a − b|k = D. Therefore, k = D/|a − b|. The larger quantity must be assigned to the larger ratio term.

Example: Two numbers are in the ratio 7:4 and their difference is 24. Thus, 3k = 24, so k = 8. The numbers are 56 and 32.

Combining Two Ratios with a Common Term

Make the common term equal in both ratios, then read the three-term ratio.

Suppose A:B = p:q and B:C = r:s. Choose multipliers so that q and r become equal. If q and r are made equal to L, the combined ratio is A:B:C = pL/q : L : sL/r.

Example: For A:B = 3:4 and B:C = 5:6, the least common multiple of 4 and 5 is 20. The ratios become A:B = 15:20 and B:C = 20:24. Hence, A:B:C = 15:20:24.

Using K in Direct and Inverse Proportion

In direct proportion use y = kx; in inverse proportion use xy = k.

For direct proportion, equal values of y/x give the same k. For inverse proportion, equal products xy give the same k. The multiplier is not used in the same way for both cases.

Example: If 5 pens cost ₹40, the direct proportionality constant is k = 40/5 = 8, so 9 pens cost 9 × 8 = ₹72. If 6 workers complete a task in 10 days, then 6 × 10 = 60 worker-days; 12 workers take 60/12 = 5 days.

K Method in Ratio and Proportion Data Table

Use the structured data below while following the explanation and examples.

K Method Form Selection

Use the relation that matches the information given in the question.

Given information K Method form Value of k
x:y = a:b x = ak, y = bk Common multiplier
x + y = T (a + b)k = T T ÷ (a + b)
Difference = D |a − b|k = D D ÷ |a − b|
Direct proportion y = kx y ÷ x
Inverse proportion xy = k x × y

Note: All quantities must use compatible units before applying the equations.

Practice K Method in Ratio and Proportion Questions

Practise published questions related to this topic.

1An article was sold at ₹1260 with a 5% loss. What was the cost price?→ 2(0.25 × 0.25 + 0.015)/(0.2 × 0.2 + 0.01) = ?→ 3Two teammates took part in a quiz contest. One teammate scored 20 points more than the other. The higher score was 70% of the combined total of both their scores. What is the sum of their scores ?→ 4If 30% of a number x equals 180, find x.→ 5Which of the following numbers is divisible by 11?→ 6A person sells an article at the profit of 10 percent. Cost price and selling price both are decreased by 20 percent. What will be the new profit percentage?→ 7The difference between cost price and selling price of an article is Rs. 800. Loss percentage is 25 percent. What is the selling price of the article?→ 863 + 41 - 29 + (33 of 1) - 18 + 26 ÷ 2 = ?→ 9If 30 percent of a number is 90, then what will be the 120 percent of that number?→ 10Simran bought an article and spent ₹777 on its repairs. She then sold it to Bharti at a profit of 15%. Bharti sold it to Charu at a loss of 20%. Charu finally sold it for ₹4600 at a profit of 25%. How much did Simran pay (in ₹) for the article.→

K Method in Ratio and Proportion Quick Quiz

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Quick Revision Notes

K Method in Ratio and Proportion: Revision Points

Use these formulas and rules for quick revision.

  • If x:y = a:b, write x = ak and y = bk.
  • For a given total T, k = T ÷ (a + b).
  • For a given difference D, k = D ÷ |a − b|.
  • For a:b:c, write the quantities as ak, bk and ck.
  • Equalise the common term before combining two ratios.
  • Direct proportion has the form y = kx.
  • Inverse proportion has the form xy = k.
  • Keep units consistent before calculating k.

Related Topic Explainers

Partnership with Change in Investment → Equivalent Ratios → Comparing Ratios → Scaling Ratios →

K Method in Ratio and Proportion FAQs

What is the formula for the K Method in Ratio and Proportion?

If x:y = a:b, then x = ak and y = bk, where k is the common multiplier.

How is k found when the sum of two quantities is given?

If the ratio is a:b and the total is T, use k = T/(a + b). For a ratio of 2:3 and total 50, k = 50/5 = 10.

How is k found when the difference between two quantities is given?

Use k = Difference/|a − b|. For a ratio of 8:5 and difference 18, k = 18/3 = 6.

How do you combine A:B = 2:3 and B:C = 4:5?

Equalise B: 2:3 becomes 8:12 and 4:5 becomes 12:15. Therefore, A:B:C = 8:12:15.

Can the K Method be used for three quantities?

Yes. If x:y:z = a:b:c, write x = ak, y = bk and z = ck. If the total is T, then k = T/(a + b + c).

What is the difference between k in direct and inverse proportion?

For direct proportion, k = y/x and y = kx. For inverse proportion, k = xy and y = k/x.

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