Question

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Answer 1

$1) \\ \frac{6m - 3}{2m} = - 3 \\ 6m - 3 = - 3 \times 2m \\ 6m - 3 = - 6m \\ 6m + 6m = 3 \\ 12m = 3 \\ m = \frac{3}{12} \\ m = \frac{1}{4} \\ \\ 2) \\ \frac{7y}{3 - 6y} = \frac{12}{5} \\ 7y \times 5 = 12 \times (3 - 6y) \\ 35y = 36 - 72y \\ 35y + 72y = 36 \\ 107y = 36 \\ y = \frac{36}{107} \\ \\ 3) \\ \frac{6t - 4}{3t - 2} = \frac{ - 7}{2} \\ 2 \times (6t - 4) = - 7 \times (3t - 2) \\ 12t - 8 = - 21t + 14 \\ 12t + 21t = 14 + 8 \\ 33t = 22 \\ t = \frac{22}{33} \\ t = \frac{2}{3} \\ \\ 4) \\ \frac{3z - 7}{ - 2z + 9} = \frac{ - 6}{11} \\ 11 \times (3z - 7) = - 6 \times ( - 2z + 9) \\ 33z - 77 = 12z - 54 \\ 33z - 12z = 77 - 54 \\ 21z = 23 \\ z = \frac{23}{21}$

5) The ratio of the ages of A and B is 9:4. Three years ago, the ratio of their ages was 3:1. Find their present ages.
solution:

A:B = 9:4

$the \: ratio \: of \: the \: present \: ages \: of \: A \: and \: B \: is \: 9 : 4 \\ \frac{A}{B} = \frac{9}{4}..........(1) \\ Three \: years \: ago, their \: ratio \: was \: 3:1\\ so, A \: : B \: = 3x : x \\ add \: 3 \: to \: both \: numerator \: and \: denominator \: to \: change \: past \: 3 \: years \: to \: present \\ \: A \: : \: B \: = \: (3x+3) \: : \: (x+3) \\ \frac{A}{B} = \frac{3x + 3}{x + 3} ...............(2) \\ equate \: (1) \: and \: (2) \\ \frac{9}{4} = \frac{3x + 3}{x + 3} \\ 9 \times (x + 3) = 4 \times (3x + 3) \\ 9x + 27 = 12x + 12 \\ 27 - 12 = 12x - 9x \\ 15 = 3x \\ x = \frac{15}{3} \\ x = 5 \\ \\ the \: present \: age \: of \: A \: is \: 3x + 3 \\so \: A=( 3 \times 5) + 3 = 15 + 3 = 18 \\ \\ the \: present \: age \: of \: B \: is \: x + 3 \\ B = 5 + 3 = 8$

6)
The numerator of a rational number is smaller than it's denominator by 3. If the numerator is decreased by 1 and the denominator is increased by 2, the number becomes 1/3. Find the rational number?

solution:

$let \: numerator \: be \: x \: and \: denominator \: be \: y \\ The \: numerator \: of \: a \: rational \: number \: is \: smaller \: than \: it's \: denominator \: by \: 3 \\ so, \: x = y - 3...........(1) \\ If \: the \: numerator \: is \: decreased \: by \: 1 \: and \: the \: denominator \: is \: increased \: by \: 2, \: the \: number \: becomes \: \frac{1}{3} \\ so, \: \frac{x - 1}{y + 2}= \frac{1}{3} ............(2) \\ substitute (1) in (2) \\ \frac{(y - 3) - 1}{y + 2} = \frac{1}{3} \\ \frac{y - 4}{y + 2} = \frac{1}{3} \\ 3 \times (y - 4) = y + 2 \\ 3y - 12 = y + 2 \\ 3y - y = 12 + 2 \\ 2y = 14 \\ y = \frac{14}{2} \\ y = 7 \\ then \: x = y - 3 = 7 - 3 = 4 \\ \\ the \: rational \: number \: is \: \frac{x}{y} = \frac{4}{7}$

7)
A streamer goes downstream and covers a certain distance in 3 hours

so, Distance be d

Time of downstream, Tds = 3hrs

it covers same distance in upstream in 5 hours

so, distance covered in upstream = distance covered in downstream = d

so, Time of upstream, Tus=5hrs

if the speed of the stream is 3 kmph

so, Speed of stream, Ss = 3 kmph

Speed of streamer in still water, Ssrw = y kmph

Speed of downstream, Sds = (Speed of streamer in still water) + (Speed of stream) = 3+y

Distance of downstream = Speed of downstream × Time for downstream
d= (3+y)×3............(1)

Speed of upstream, Sus = (Speed of streamer in still water) - (Speed of stream)
d= (y-3)×5............(2)

equate (1) and (2)
(3+y)3 = (y-3)5
9+3y = 5y - 15
9+15 = 5y - 3y
2y = 24
y = 24/2
y= 12

so, Speed of streamer in still water, Ssrw = y = 12 kmph

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