Question

1. Two numbers are in the ratio 7: 11. If 7 is added to each of the numbers, the ratio becomes 2: 3. Find the numbers.


2. What number must be added to each term of the ratio 7 : 12 to make it 2 : 3​

Answer 1

$\bf \underline{ \underline{\maltese\:Given} }$

$\sf Two \: numbers \: are \: in \: the \: ratio \: 7: 11$

$\sf If \: 7 \: is \: added \: to \: each \: of \: the \: numbers :$

$\sf \implies \bf{ \underline{Then}}, \sf the \: ratio \: becomes \: 2: 3$

$\bf \underline{ \underline{\maltese\:To \: find} }$

$\sf \implies Both \: the \: numbers = \: ?$

$\bf \underline{ \underline{\maltese\:Solution} }$

$\sf Let \: us \: assume \: that,$

$\sf \implies First \: number \: be \: 7x$

$\sf \implies Second \: number \: be \: 11x$

$\sf A ccording \: to \: questions,$

$\sf \implies Equation = \bf \red{ \dfrac{7x + 7}{11x + 7} = \dfrac{2}{3} }$

$\sf Doing \: cross \: multiplication,$

$\sf \implies 3(7x + 7) = 2(11x + 7)$

$\sf \implies 21x + 21 = 22x + 14$

$\sf \implies 21x + 21 - 22x = 14$

$\sf \implies - x + 21 = 14$

$\sf \implies - x = 14 - 21$

$\sf \implies - x = - 7$

$\bf \implies x = 7$

$\sf \bigstar \: The \: value \: of \: x \: is \: 7$

$\bf \underline{ Therefore},$

$\sf \implies First \: number \: (7x) = \bf 7 \times 7 = 49$

$\sf \implies Second \: number \: (11x) = \bf 11 \times 7 = 77$

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$\bf \underline{ \underline{\maltese\:Given} }$

$\sf The \: ratio \: of \: two \: numbers = 7 : 13$

$\sf After \: adding \: 1 \: term, then \: the \: ratio \: becomes \: 2 : 3$

$\bf \underline{ \underline{\maltese\:To \: find} }$

$\sf \implies The \: term = \: ?$

$\bf \underline{ \underline{\maltese\:Solution} }$

$\sf \implies Let \: the \: term \: to \: be \: added \: be \: x$

$\bf \underline{ Then},$

$\sf \implies Equation = \bf \red{ \dfrac{7 + x}{12 + x} = \dfrac{2}{3} }$

$\sf Doing \: cross \: multiplication,$

$\sf \implies 3(7 + x) = 2(12 + x )$

$\sf \implies 21 + 3x = 24 + 2x$

$\sf \implies 21 + 3x - 2x = 24$

$\sf \implies 21 + x = 24$

$\sf \implies x = 24 - 21$

$\sf \implies x = 3$

$\sf \bigstar \: The \: value \: of \: x \: is \: 3$

$\underline{ \boxed{ \red{ \bf Therefore, \: number \: to \: be \: added \: is \: 3 }}}$