Solution: (c)

Let monthly income of Sneha, Tina and Akruti is 95x, 110x and 116x respectively

Annual income of Sneha = 12 \(\displaystyle \times \) 95x = 3,42,000

X = \(\displaystyle \frac{{342000}}{{95\times 12}}\)

Annual income of Akruti = 12 \(\displaystyle \times \) 116x

= \(\displaystyle 12\times 116\times \frac{{342000}}{{95\times 12}}\)

Akruti’s annual income = ₹ 4,17,600

Alternate method

Irrespective of monthly or annual income, we can solve the question as given below:

Sneha’s Income / Akruti Income = 95/116

342000 / Akruti Income=95/116

Akruti Income= (116 \(\displaystyle \times \) 342000)/95 = 417600

Solution: (d)

Ratio of monthly income and ratio of annual income will be the same, ie 84 : 76 : 89

Therefore, Sum of Sita’s and Kunal’s annual income

= \(\displaystyle \frac{{456000}}{{76}}\times (84+89)=1038000\)

Alternate method

Let monthly income of Sita Riya and Kunal be 84k, 76k and 89k, respectively.
Given annual income of Riya = 456000
Therefore, Monthly income of Riya = 456000/12 = 38000
\(\displaystyle \Rightarrow \) 76k = 38000

\(\displaystyle \Rightarrow \) k = 500
So, the monthly income of Sita and Kunal = 84k + 89k = 173k
= 173 \(\displaystyle \times \) 500 = 86500
Therefore, annual income = 86500 \(\displaystyle \times \) 12 = ₹ 1038000

Solution: (b)

\(\displaystyle \frac{{x+1}}{{y+2}}=\frac{2}{3}\)\(\displaystyle \Rightarrow 3x-2y=1\)

\(\displaystyle \frac{{x+2}}{{y+3}}=\frac{5}{7}\) \(\displaystyle \Rightarrow 7x-5y=1\)

Or, \(\displaystyle 3x-2y=7x-5y\)

\(\displaystyle \Rightarrow 3y=4x\)

\(\displaystyle \Rightarrow \)\(\displaystyle \frac{x}{y}=\frac{3}{4}\)

Solution: (e)

According to the question

Present age of Parineeta = 33 – 9 = 24 years

Present age of Manisha = 24 – 9 = 15 years

Present age of Deepali = 24 + 15 = 39 years

Therefore, 5 : X = 15 : 39

\(\displaystyle \Rightarrow \)x = \(\displaystyle \frac{{5\times 39}}{{15}}=13\)

Alternate method

Present age of Parineeta is 33 – 9 = 24 years

Age of Manisha = 24 – 9 = 15 years

But from the given information, difference Mamsha and Deepali’s age is 24 years

Deepali’s Age =15+24=39 years

Thus, Deepali’s present age is 39 years.

Now, as the ratio of Manisha’s age and Deepali’s age is , so…

Manisha / Deepali=5 / x

15/39=5/x

15x=95

x=195/15

Therefore, x=13

So, the value of X is 13

Solution: (a)

Let Gloria’s and Sara’s present ages be 4x and 7x years respectively.

Two years ago, \(\displaystyle \frac{{4x-2}}{{7x-2}}=\frac{1}{2}\)

\(\displaystyle \Rightarrow \) 8x – 4 = 7x – 2

\(\displaystyle \Rightarrow \) x= 2

Therefore, Sara’s age three years hence = 7x + 3 = 17 years

Alternate method

Let the present age of Gloria and Sara be A and B respectively

It is given that, \(\displaystyle \frac{A}{B}=\frac{4}{7}\)———–(1)

Two Years back Gloria age will be A – 2 and Sara age will be B – 2 and which is given that

\(\displaystyle \frac{{A-2}}{{B-2}}=\frac{1}{2}\)————(2)

Solving the two equations we get B=14 years, so three years hence age of Sara will be=14+3=17 years

Solution: (e)

Let present age of father, mother and daughter be

7x, 6x, 2x

Given, 6x – 2x = 24

\(\displaystyle \Rightarrow \)4x = 24

\(\displaystyle \Rightarrow \) x = 6

Father age = 7x = 42 years.

Solution: (e)

9, 15, 27

9 – x , 15 – x, 27 – x

\(\displaystyle \frac{{15-x}}{{9-x}}=\frac{{27-x}}{{15-x}}\)

\(\displaystyle {{(15-x)}^{2}}=(27-x)(9-x)\)

\(\displaystyle 225+{{x}^{2}}-30x=243-9x-27x+{{x}^{2}}\)

\(\displaystyle \Rightarrow \)  –30x + 9x + 27x = 243 –225

\(\displaystyle \Rightarrow \) 6x = 18

\(\displaystyle \Rightarrow \) x = 3

Alternate method:

If, \(\displaystyle \frac{a}{b}=\frac{b}{c}\) be the continued fraction

so, \(\displaystyle \frac{{9-x}}{{15-x}}=\frac{{15-x}}{{27-x}}\)

[9, 15 and 27 are divisible by 3]

We can see that, x = 3, satisfies the above condition.

\(\displaystyle \frac{{9-3}}{{15-3}}=\frac{{15-3}}{{27-3}}\)

\(\displaystyle \Rightarrow \frac{6}{{12}}=\frac{{12}}{{24}}\)

\(\displaystyle \Rightarrow \frac{1}{2}=\frac{1}{2}\)

Therefore, x=3