Two number are in the ratio \(3 : 5.\) If \(9\) is subtracted from each, the new numbers are in the ratio \(12 : 23.\) The smaller number is:
Answer: B
Let the numbers be \(3x\) and \(5x.\)
Then, \(\frac{3x - 9}{5x - 9} = \frac{12}{23}\)
\(\Rightarrow 23(3x - 9) = 12(5x - 9)\)
\(\Rightarrow 9x = 99\)
\(\Rightarrow x = 11.\)
\(\therefore\) The smaller number \(= (3 \times 11) = 33.\)
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The fourth proportional to \(5, 8, 15\) is:
Answer: D
Let the fourth proportional to \(5, 8 , 15\) be \(x.\)
Then, \(5 : 8 : 15 : x\)
\(\Rightarrow 5x = (8 \times 15)\)
\(x = \frac{(8 \times 15)}{5} = 24.\)
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Two numbers are respectively \(20\)% and \(50\)% more than a \(3rd\) number. The ratio of the two numbers is:
Answer: B
Let the \(3rd\) number be \(x.\)
Then, first number \(= 120\)% of \(x = \frac{120x}{100} = \frac{6x}{5}\)
Second number \(= 150 \)% of \(x = \frac{150x}{100} = \frac{3x}{2}\)
\(\therefore\) Ratio of first two numbers \(= (\frac{6x}{5} : \frac{3x}{2}) = 12x : 15x = 4:5\)
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The ratio of the number of boys and girls in a college is \(7 : 8.\) If the percentage increase in the number of boys and girls be \(20\)% and \(10\)% respectively, what will be the new ratio?
Answer: A
Let the number of boyes and girls in the collage be \(7x\) and \(8x\) respectively. Their increased number is \((120\)% of \(7x)\) and \((110\)% of \(8x)\).
\(\Rightarrow (\frac{120}{100} \times 7x)\) and \((\frac{110}{100} \times 8x)\)
\(\Rightarrow \frac{42x}{5} \) and \(\frac{44x}{5}\)
\(\therefore\) The required ratio \(= (\frac{42x}{5} : \frac{44x}{5}) = 21 :22.\)
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Seats for Mathematics, Physics and Biology in a school are in the ratio \( 5 : 7 : 8.\) There is a proposal to increase these seats by \(40\)%, \(50\)% and \(75\)% respectively. What will be the ratio of increased seats?
Answer: C
Let the number of seats for Mathematics, Physics and Biology be \(5x, 7x\) and \(8x\)
Number of increased seats are \((140\)% of \(5x),\) \((150\)% of \(7x)\) and \((175\)% of \(8x).\)
\(\Rightarrow (\frac{140}{100} \times 5x), (\frac{150}{100} \times 7x)\) and \((\frac{175}{100} \times 8x)\)
\(\Rightarrow 7x, \frac{21x}{2}\) and \(14x.\)
\(\therefore\) The required ratio \(= 7x : \frac{21x}{2} : 14x\)
\(\Rightarrow 14x : 21x : 28x\)
\(\Rightarrow 2 : 3 : 4.\)
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If \(Rs. 782\) be divided into three parts, proportional to \(\frac{1}{2} : \frac{2}{3} : \frac{3}{4}, \) then the first part is:
Answer: D
Given ratio \(= \frac{1}{2} :\frac{2}{3} : \frac{3}{4} = 6:8:9\)
\(\therefore \) 1st part \(= Rs. (782 \times \frac{6}{23}) = Rs. 204\)
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A sum of money is to be distributed among \(A, B, C, D\) in the proportion of \(5 : 2 : 4 : 3.\) If \(C\) gets \(Rs. 1000\) more than \(D,\) what is \(B's\) share?
Answer: A
Let the shares of \(A, B, C\) and \(D\) be \(RS. 5x, Rs. 2x, Rs. 4x\) and \(Rs. 3x\) respectively.
Then, \(4x - 3x = 1000\)
\(\Rightarrow x = 1000.\)
\(\therefore B's\) share \(= Rs. 2x = Rs. (2 \times 1000) = Rs. 2000.\)
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The salaries \(A, B, C\) are in the ratio \(2 : 3 : 5.\) If the increments of \(15\)%, \(10\)% and \(20\)% are allowed respectively in their salaries, then what will be new ratio of their salaries?
Answer: C
Let \(A = 2k, B = 3k\) and \(C = 5k.\)
\(A's\) new salary \(= \frac{115}{100}\) of \(2k = (\frac{115}{100} \times 2k) = \frac{23k}{10}\)
\(B's\) new salary \(= \frac{110}{100}\) of \(3k = (\frac{110}{100} \times 3k) = \frac{33k}{10}\)
\(C's\) new salary \(= \frac{120}{100}\) of \(5k = (\frac{120}{100} \times 5k) = 6k\)
\(\therefore\) New ratio \((\frac{23k}{10} : \frac{33k}{10} : 6k) = 23 : 33 : 60\)
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Salaries of Piu and Amit are in the ratio \(2 : 3.\) If the salary of each is increased by \(Rs. 4000,\) the new ratio becomes \(40 : 57.\) What is Amit's salary?
Answer: A
Let the original salaries of Piu and Amit be \(Rs. 2x\) and \(Rs. 3x\) respectively.
Then, \(\frac{2x + 4000}{3x + 4000} = \frac{40}{57}\)
\(\Rightarrow 57(2x + 4000) = 40(3x + 4000)\)
\(\Rightarrow 6x = 68,000\)
\(\Rightarrow 3x = 34,000\)
Amit's presents salary \(= (3x + 4000) = Rs.(34000 + 4000) = Rs. 38,000.\)
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The sum of three numbers is \(98.\) If the ratio of the first to second is \(2 :3\) and that of the second to the third is \(5 : 8,\) then the second number is:
Answer: B
Let the three part be \(A, B, C.\)
Then, \(A:B = 2:3\) and \(B:C = 5:8 = (5 \times \frac{3}{5}):(8 \times \frac{3}{5}) = 3:\frac{24}{5}\)
\(\Rightarrow A:B:C = 2:3:\frac{24}{5} =10:15:24\)
\(\Rightarrow B = (98 \times \frac{15}{49}) = 30.\)
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