# Ratio and Propotion

$m,~ n,~p,~q,~are~real ~numbers~,p~and~q~ both~ together~ not ~equal~to~zero. \\ If~~\dfrac {a}{b} =\dfrac {c}{d} \\ \implies \dfrac {a}{b} =\dfrac {c}{d}=\dfrac {pa+qc}{pb+qd}.\\Also\dfrac {ma+nb}{pa+qb} =\dfrac {mc+nd}{pc+qd}$
This can be applied any where with a, b, c, d, are any expression. We can apply it in solving simultaneous equations in two variables. It is similar to elimination method.
$2a - 5b = 11$
$3a+2b = 7$
$\implies \dfrac{2a - 5b }{3a+2b } = \dfrac{11}{7}.~~~~\$/extract_itex] Subtract numerator from denominator on both the sides. $\implies \dfrac{2a - 5b }{a+7b } = \dfrac{11}{-4}$ Subtract twice the denominator from the numerator on both the sides. $\implies \dfrac{-19b}{a+7b } = \dfrac{19}{-4}~\\$ $\implies \dfrac{b }{a+7b } = \dfrac{1}{4}~~~~~~~~~~~~\\$ Subtract seven times the numerator from denominator on both the sides. $\implies \dfrac{b }{a} = \dfrac{1}{-3}$ $\implies a=-3b~\\$ Substitute in any one equation and we get ..... a=3~~~b=-1 Note by Niranjan Khanderia 5 years, 3 months ago This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science. When posting on Brilliant: • Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused . • Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone. • Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge. • Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events. MarkdownAppears as *italics* or _italics_ italics **bold** or __bold__ bold - bulleted- list • bulleted • list 1. numbered2. list 1. numbered 2. list Note: you must add a full line of space before and after lists for them to show up correctly paragraph 1paragraph 2 paragraph 1 paragraph 2 [example link](https://brilliant.org)example link > This is a quote This is a quote  # I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world" # I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world" MathAppears as Remember to wrap math in $$ ... $$ or \[ ... $ to ensure proper formatting.
2 \times 3 $2 \times 3$
2^{34} $2^{34}$
a_{i-1} $a_{i-1}$
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\sum_{i=1}^3 $\sum_{i=1}^3$
\sin \theta $\sin \theta$
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i think above method is lengthy for such a trivial problem of simultaneous equations

- 3 years, 6 months ago

It was simply to show how the method is used in short. If I gave where it is actually used , explanation would be long.

- 3 years, 4 months ago