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# Given Four Quantities P, Q, R and S Are in Proportion, Show that Q2(P-r):Rs(Q-s)=(P2-q2-pq): ( R2-s2-rs). - ICSE Class 10 - Mathematics

#### Question

Given four quantities p, q, r and s are in proportion, show that
q2(p - r) : rs (q - s) =(p2- q2- pq): ( r2-s2-rs).

#### Solution

p, q, r and s are 1n proportion

then, p : q :: r : s

Let "p"/"q" = "r"/"s" = "k"

Then p = kq and r =ks

Now, we have to prove that

(("p" - "r")"q"^2)/(("q - s")"rs") = ("p"^2 - "q"^2 - "pq")/("r"^2 - "s"^2 - "rs")

LHS

= (("p" - "r")"q"^2)/(("q - s")"rs")

= (("kq" - "ks")"q"^2)/(("q - s")"ks" xx "s")

= ("k"("q - s")"q"^2)/("ks"^2 ("q - s"))

= "q"^2/"s"^2

RHS

= ("p"^2 - "q"^2 - "pq")/("r"^2 - "s"^2 - "rs")

= ("k"^2"q"^2 - "q"^2 - "kq" xx "q")/("k"^2"s"^2 - "s"^2 - "ks" xx "s")

= ("q"^2("k"^2 - 1 - "k"))/("s"^2("k"^2 - 1 - "k"))

= "q"^2/"s"^2

LHS = RHS

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#### APPEARS IN

Frank Solution for Frank Class 10 Mathematics Part 2 (2016 to Current)
Chapter 9: Ratio and Proportion
Exercise 9.2 | Q: 10

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Solution Given Four Quantities P, Q, R and S Are in Proportion, Show that Q2(P-r):Rs(Q-s)=(P2-q2-pq): ( R2-s2-rs). Concept: Proportions.
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