Equations 1/4

19. In an examination, three marks will be awarded for every correct answer and one mark will be deducted for every wrong answer. A student attempted 70 questions and scored 170 marks. Find the number of questions he answered correctly.
a. 55
b. 58
c. 60
d. 65
Explanation:
Let the number of questions he answered correctly = $c$, then wrong questions = $75-c$
$\Rightarrow 3 \times c + \left( {70 - c} \right)\times -1 = 170$
$\Rightarrow 3c - 70 + c = 170$
$\Rightarrow 4c = 240$
$\Rightarrow c = 60$

20. If 3 is added to the denominator of a fraction, it becomes $\displaystyle\frac{1}{3}$ and if 4 be added to its numerator, it becomes $\displaystyle\frac{3}{4}$ ; the fraction is :
a. $\displaystyle\frac{4}{9}$
b. $\displaystyle\frac{3}{{20}}$
c. $\displaystyle\frac{7}{{24}}$
d. $\displaystyle\frac{5}{{12}}$
Explanation:
Let the required fraction be $\displaystyle\frac{a}{b}$
Then $\displaystyle\frac{a}{{b + 3}} = \displaystyle\frac{1}{3}$ $\Rightarrow 3a - b = 3$ and
$\displaystyle\frac{{a + 4}}{b} = \displaystyle\frac{3}{4} \Rightarrow 4a - 3b = -16$
Solving, we get $a = 5, b = 12$;
Required answer = $\dfrac{5}{{12}}$

21. Of the three numbers, the first is twice the second and is half of the third. If the average of three numbers is 56, then the smallest number is
a. 24
b. 36
c. 40
d. 48
Explanation:
Let the second number is $a$. Then the first number is $2a$ and third number is $4a$.
$\displaystyle\frac{{2a + a + 4a}}{3} = 56 \Rightarrow 7a = 3 \times 56$ or
$a = \displaystyle\frac{{3 \times 56}}{7} = 24$
Smallest number is 24

22. The difference of two numbers is 8 and ${\left( {\dfrac{1}{{12}}} \right)^{th}}$ of the sum is 1. The numbers are
a. 10, 2
b. 18, 26
c. 10, 18
d. 26, 34
Explanation:
Let the numbers be $a$ and $(a+8)$. Then
$\displaystyle\frac{1}{{12}}\left( {a + (a + 8)} \right) = 1$
$\Rightarrow 2a + 8 = 12$
$\Rightarrow a = 2, a + 8 = 10$

23. A number is 25 more than its ${\left( {\frac{2}{5}} \right)^{th}}$. The number is
a. 60
b. 80
c. $\displaystyle\frac{{125}}{3}$
d. $\displaystyle\frac{{125}}{7}$
Explanation:
Let the number be N. Then
$N - \dfrac{2}{5}N = 25$
$\Rightarrow \dfrac{{3N}}{5} = 25$
$\Rightarrow N = \dfrac{{125}}{3}$

24. The sum of three numbers is 68. If the ratio between first and second be 2 : 3 and that between second and third be 5 : 3, then the second number is
a. 30
b. 20
c. 58
d. 48
Explanation:
Let the numbers be $x,y,z$.  Then
$\begin{array}{l} x:y = 2:3\\[4px] y:z = 5:3 \end{array}$
Multiply the first ratio by 5 and second ratio by 3 to equate $y$ terms.
$\begin{array}{l} x:y = \left( {2:3} \right) \times 5 = 10:15\\[4px] y:z = \left( {5:3} \right) \times 3 = 15:9 \end{array}$
$x:y:z = 10:15:9$
Let the numbers be $10k$, $15k$, $9k$
Given, $10k + 15k + 9k = 68$
$\Rightarrow k = \dfrac{{68}}{{34}} = 2$
$\therefore$ second number = $15 \times 2 = 30$

25. The sum of two numbers is 100 and their difference is 37. The difference of their squares is
a. 37
b. 100
c. 63
d. 3700
Let the numbers be $x$ and $y$
Then $x+y=100$ & $x-y=37$
${x^2} - {y^2} = (x - y)(x + y) = 100 \times 37 = 3700$