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# If (A + B + C + D) (A – B – C + D) = (A + B – C – D) (A – B + C – D), Prove that A: B = C: D. - ICSE Class 10 - Mathematics

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ConceptComponendo and Dividendo Properties

#### Question

If (a + b + c + d) (a – b – c + d) = (a + b – c – d) (a – b + c – d), prove that a: b = c: d.

#### Solution

Given (a + b + c + d)/(a – b – c + d) = (a + b – c – d)/(a – b + c – d)

Applying componendo and dividendo

((a + b + c + d) + (a + b - c - d))/((a + b + c + d) - (a + b - c - d)) = ((a - b + c - d) + (a - b - c + d))/((a - b + c - d) - (a - b - c + d))

(2(a + b))/(2(c + d)) = (2(a - b))/(2(c - d))

(a + b)/(c + d) = (a - b)/(c -d)

(a + b)/(a - b) = (c + d)/(c - d)

Applying componendo and dividendo

(a + b + a - b)/(a + b - a - b) = (c + d + c - d)/(c + d - c + d )

(2a)/(2b) = (2c)/(2d)

a/b = c/d

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#### APPEARS IN

Selina Solution for Selina ICSE Concise Mathematics for Class 10 (2018-2019) (2017 to Current)
Chapter 7: Ratio and Proportion (Including Properties and Uses)
Ex.7C | Q: 7

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Solution If (A + B + C + D) (A – B – C + D) = (A + B – C – D) (A – B + C – D), Prove that A: B = C: D. Concept: Componendo and Dividendo Properties.
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