Componendo and Dividendo: Formula, Rules and Examples

Componendo and Dividendo is a ratio theorem used to transform an equation of two equal ratios. It combines the corresponding terms by addition and subtraction to form a new ratio. The method helps solve ratio theorem questions involving expressions such as (a + b)/(a - b), while keeping the equality valid under the required non-zero denominator conditions.

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What is Componendo and Dividendo?

If a/b = c/d, then componendo and dividendo gives (a + b)/(a - b) = (c + d)/(c - d), provided the denominators are non-zero.

The theorem follows from the separate operations: componendo adds the denominator to the numerator, while dividendo subtracts the denominator from the numerator. Thus, from a/b = c/d, we get (a + b)/b = (c + d)/d and (a - b)/b = (c - d)/d. Dividing these two results gives the combined form. For example, if a/b = 3/2, then (a + b)/(a - b) = (3 + 2)/(3 - 2) = 5.

Componendo and Dividendo Formula & Tricks

Important Formulas

Componendo
If a/b = c/d, then (a + b)/b = (c + d)/d

Add each denominator to its corresponding numerator.

Dividendo
If a/b = c/d, then (a - b)/b = (c - d)/d

Subtract each denominator from its corresponding numerator.

Componendo and Dividendo
If a/b = c/d, then (a + b)/(a - b) = (c + d)/(c - d)

Apply componendo and dividendo together; the displayed denominators must be non-zero.

Reverse ratio form
If (x + y)/(x - y) = m/n, then x/y = (m + n)/(m - n)

Cross-multiplication gives n(x + y) = m(x - y), so (m - n)x = (m + n)y.

Quick Tricks

Convert the final expression directly

When a ratio is given as (x + y)/(x - y) = m/n, write x/y = (m + n)/(m - n) instead of solving two separate equations.

Example: (x + 3)/(x - 3) = 5/2 gives x/3 = (5 + 2)/(5 - 2) = 7/3, so x = 7.
Use a common multiplier for equal ratios

If a/b = c/d, represent the terms as a = kb and c = kd, or use proportional pairs. This makes addition and subtraction transformations immediate.

Example: If a:b = 3:2, take a = 3k and b = 2k. Then (a + b):(a - b) = 5k:1k = 5:1.

Componendo and Dividendo Concepts

Separate Componendo Rule

Componendo changes a/b = c/d into (a + b)/b = (c + d)/d.

The denominator remains unchanged, and the denominator is added to the numerator on each side. This rule is valid when the original denominators b and d are non-zero.

Example: From 4/7 = 8/14, componendo gives 11/7 = 22/14.

Separate Dividendo Rule

Dividendo changes a/b = c/d into (a - b)/b = (c - d)/d.

The denominator remains unchanged, and the denominator is subtracted from the numerator on each side. The resulting denominators must also be non-zero wherever the expression is used.

Example: From 5/3 = 10/6, dividendo gives 2/3 = 4/6.

Combined Componendo and Dividendo

Applying both operations gives (a + b)/(a - b) = (c + d)/(c - d).

Starting with a/b = c/d, the separate forms are (a + b)/b = (c + d)/d and (a - b)/b = (c - d)/d. Dividing the first transformed equality by the second produces the combined theorem, provided a - b and c - d are non-zero.

Example: If a:b = 7:3, then (a + b):(a - b) = 10:4 = 5:2.

Solving an Unknown from a Transformed Ratio

For (x + y)/(x - y) = m/n, the corresponding original ratio is x/y = (m + n)/(m - n).

Cross-multiply: n(x + y) = m(x - y). Rearranging gives (m - n)x = (m + n)y, so x/y = (m + n)/(m - n). This requires m - n to be non-zero for the displayed result.

Example: If (p + q)/(p - q) = 7/3, then p/q = 10/4 = 5/2.

Componendo and Dividendo Video Lessons

Watch short topic-wise lessons for quick revision.

4 Lessons
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Duplicate, Triplicate and Sub-Duplicate Ratios

Learn the definitions and application of duplicate, triplicate, and sub-duplicate ratios, with clear methods for understanding how these special ratios are formed and used.

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Practice Componendo and Dividendo Questions

Practise published questions related to this topic.

1A shopkeeper marks his goods at X% above the cost price and sells them at a discount of 35%. If he makes a profit of 69%, find the value of X.→ 2Which of the following will have 25 percent discount?→ 3A shopkeeper has 144 chocolates, 180 toffees, and 216 candies. He wants to pack them into boxes such that each box has the same number of chocolates, toffees, and candies, and no item is left. Find the maximum number of boxes.→ 4(√0.0169/1.3) × (0.18/√3.24) × (0.24/√0.0576) = ?→ 573P2 is divisible by 8. Which of the following cannot be the value of P?→ 6The 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?→ 7A mobile phone is marked at ₹8,000. If two successive discounts of 15% and 10% are given, what is the selling price?→ 8A shopkeeper offers the following schemes on toys of the same marked price. (A) Successive discounts of 50% and 16% on any number of toys bought. (B) Successive discounts of 49%, 8% and 13% on any number of toys bought. (C) 35% discount on the first 6 toys and 45% discount on each toy thereon. (D) On buying eight items, the customer is billed for only four items. A customer wants to buy 8 toys. Which of the above schemes is the least beneficial to her?→ 9Cost price of 15 articles is Rs. 197. Selling price of 12 articles is Rs. 197. What is the ratio of the value of profit/loss of 1 article to the selling price of 1 article? (Cost price of all the articles is same. Selling price of all the articles is same.)→ 10Least Common Multiple of A and B is 15. Least Common Multiple of C and D is 25. What is the Least Common Multiple of A, B, C and D?→

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Quick Revision Notes

Componendo and Dividendo Revision Points

Use these direct rules and conditions while solving ratio equations.

  • From a/b = c/d, componendo gives (a + b)/b = (c + d)/d.
  • From a/b = c/d, dividendo gives (a - b)/b = (c - d)/d.
  • The combined result is (a + b)/(a - b) = (c + d)/(c - d).
  • For (x + y)/(x - y) = m/n, use x/y = (m + n)/(m - n).
  • Check that every denominator in the original and transformed expression is non-zero.
  • The theorem applies to equal ratios, not to ratios that are merely added or subtracted across the equality.

Componendo and Dividendo FAQs

What is the componendo and dividendo formula?

If a/b = c/d, then (a + b)/(a - b) = (c + d)/(c - d), provided a - b and c - d are non-zero.

What is the difference between componendo and dividendo?

Componendo adds the denominator to the numerator: (a + b)/b. Dividendo subtracts the denominator from the numerator: (a - b)/b.

If (x + 2)/(x - 2) = 3, what is x?

Write 3 as 3/1. Then x/2 = (3 + 1)/(3 - 1) = 4/2 = 2, so x = 4.

If a:b = 5:3, find (a + b):(a - b).

Using the ratio parts, (a + b):(a - b) = (5 + 3):(5 - 3) = 8:2 = 4:1.

Can componendo and dividendo be applied when the numerator equals the denominator?

Not in the combined form, because a - b becomes zero and the resulting ratio is undefined. The original ratio may still be defined if b is non-zero.

How do you find x/y from (x + y)/(x - y) = 4/3?

Use x/y = (4 + 3)/(4 - 3) = 7/1. Therefore, x:y = 7:1.

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