Ratio and Proportion - 02

Explanation:-

Let the age of A & B = 5x and 6x years
According to question, $$\eqalign{ & \frac{{5x + 7}}{{6x + 7}} = \frac{6}{7} \cr & 35x + 49 = 36x + 42 \cr & x = 7 \cr & {\text{A's Present age}} = 5x \cr & = 5 \times 7 = {\text{35 years}} \cr}$$

Explanation:-

Let,
⇒ Sonali's age = 5x
⇒ Monali's age = 3x
According to the question, $$\eqalign{ & \Rightarrow \frac{{5x + 5}}{{3x + 5}} = \frac{{10}}{7} \cr & \Rightarrow \frac{{x + 1}}{{3x + 5}} = \frac{2}{7} \cr & \Rightarrow 7x + 7 = 6x + 10 \cr & \Rightarrow x = 3 \cr}$$ ⇒ So, Monali's present age = 3x
= 3 × 3 = 9 years

Explanation:-

Let the number of ladies and gents at the party be 3x and 2x respectively.
Then, $$\eqalign{ & = \frac{{3x}}{{2x + 20}} = \frac{2}{3} \cr & \Rightarrow 9x = 4x + 40 \cr & \Rightarrow 5x = 40 \cr & \Rightarrow x = 8. \cr}$$ ∴ Number of ladies = 3 × 8 = 24

Explanation:-

Let the number to be added be x
Then, $$\eqalign{ & = \frac{{3 + x}}{{5 + x}} = \frac{5}{6} \cr & \Rightarrow 6\left( {3 + x} \right) = 5\left( {5 + x} \right) \cr & \Rightarrow 18 + 6x = 25 + 5x \cr & \Rightarrow x = 7. \cr}$$

Explanation:-

Let the number of managers and production - line workers be 5x and 72x respectively. Then, $$\eqalign{ & = \frac{{5x}}{{72x + 8}} = \frac{5}{{74}} \cr & \Rightarrow 370x = 360x + 40 \cr & \Rightarrow 10x = 40 \cr & \Rightarrow x = 4. \cr}$$ ∴ Number of managers = 5 × 4 = 20

Explanation:-

$$\eqalign{ & {\text{Quantity of milk}} \cr & = \left( {80 \times \frac{7}{{10}}} \right){\text{litres}} \cr & = {\text{56 litres}}{\text{.}} \cr & {\text{Quantity of water}} \cr & = \left( {80 - 56} \right){\text{litres}} \cr & = {\text{24 litres}}{\text{.}} \cr}$$ Let the quantity of water to be added be x litres.
Then, $$\eqalign{ & = \frac{{56}}{{24 + x}} = \frac{2}{1} \cr & \Rightarrow 2x = 8 \cr & \Rightarrow x = 4. \cr}$$

Explanation:-

Let the required number be x
Then, $$= \frac{{7 - x}}{{9 - x}} = \frac{{11 - x}}{{15 - x}}$$ ⇒ (7 - x)(15 - x) = (11 - x)(9 - x)
⇒ 105 - 22x + x² = 99 - 20x + x²
⇒ 2x = 6
⇒ x = 3

Explanation:-

According to the question, $$\eqalign{ & \frac{{{\text{Story books}}}}{{{\text{Other books}}}} = \frac{7}{2} \cr & {\text{Story books = 1512}} \cr & {\text{7 units}} \to 1{\text{512}} \cr & {\text{1 unit}} \to \frac{{1512}}{7} = 216 \cr & {\text{2 units}} \to 216 \times 2 = 432 \cr & {\text{Other books }} = 432 \cr & {\text{New ratio of}} \cr & \frac{{{\text{Story books}}}}{{{\text{Other books}}}} = \frac{{15}}{4} \cr}$$ As we know that only story books are added
$$\eqalign{ & {\text{4 units}} \to 43{\text{2}} \cr & {\text{1 unit}} \to \frac{{432}}{4} = 108 \cr & {\text{15 units}} \to 108 \times 15 \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 1620 \cr}$$ New collection of story books = 1620
∴ NUmber of story books are added = 1620 - 1512 = 108

Explanation:-

Let the basic salaries of A and B be Rs. x and Rs. y respectively.
Then,
= x + 65% of x = y + 80% of y $$\eqalign{ & \Rightarrow \frac{{165}}{{100}}x = \frac{{180}}{{100}}y \cr & \Rightarrow 165x = 180y \cr & \Rightarrow \frac{x}{y} = \frac{{180}}{{165}} = \frac{{12}}{{11}} \cr & \therefore {\text{Required ratio}} \cr & = 12:11 \cr}$$

Explanation:-

	Rs. 1	:	50 P	:	25 p
Values of coins	8x	:	11x	:	3x
Number of coins	8x×1	:	4x×2	:	3x×4
	8x	:	8x	:	12x

$$\eqalign{ & \therefore {\text{Total coins}} \cr & \Rightarrow 8x + 8x + 12x = 28x \cr & \Rightarrow 28x = 280{\text{ (given)}} \cr & \Rightarrow x = \frac{{280}}{{28}} \cr & \Rightarrow x = 10 \cr}$$ ∴ Number of 50 P coins are 8x
= 8 × 10 = 80