Question

A two-digit number is such that product of its digits is 35. If 18 is added to the number, the digits interchange their places. Find the number.

Or, two water taps together can fill a tank in $9\frac{3}{8}$ hours. The larger tap takes 10 hours less than the smaller one to fill the tank separately. Find the time in which each tap can separately fill the tank.

Answer 1

Let the tens digit be x and the units digit be y.
∴ Product of the digits = xy = 35............(i)
Original number is (10x + y).
If the digits interchange their position, the number becomes (10y + x).
Thus,
(10x + y) + 18 = (10y + x)
9x − 9y = − 18
x − y = − 2 .............(ii)
From (i), we get: $y=\frac{35}{x}$
On substituting the value of y in (ii), we get:
$x-\frac{35}{x}=-2$
$\Rightarrow x^2-35=-2x\Rightarrow x^2+2x-35=0$
$\Rightarrow x^2+7x-5x-35=0$
$\Rightarrow x\left(x+7\right)-5\left(x+7\right)=0$
$\Rightarrow \left(x-5\right)\left(x+7\right)=0$
⇒ x = 5 or x = − 7
Since the digits cannot be negative, x = 5 and $y=\frac{35}{5}=7$
Hence, the required number = 10(5) + 7 = 57

OR

Let the smaller tap fill the tank in x hours.
Then, the larger tap will fill it in (x − 10) hours.
Time taken by both together to fill the tank = $9\frac{3}{8}=\frac{75}{8}$ hours
Part filled by smaller tap in 1 hour = $\frac{1}{x}$
Part filled by larger tap in 1 hour = $\frac{1}{\left(x-10\right)}$
Part filled by both the taps in 1 hour = $\frac{8}{75}$
∴ $\frac{1}{x}+\frac{1}{\left(x-10\right)}=\frac{8}{75}$
$\Rightarrow \frac{\left(x-10\right)+x}{x\left(x-10\right)}=\frac{8}{75}\Rightarrow \frac{\left(2x-10\right)}{x\left(x-10\right)}=\frac{8}{75}$
75(2x −10) = 8x(x −10) [By cross−multiplication]
⇒ 150x − 750 = 8x² − 80x
⇒ 8x² − 230x + 750 = 0
⇒ 4x² − 115x + 375 = 0
⇒ 4x² − 100x − 15x + 375 = 0 ⇒ 4x(x − 25) − 15(x − 25) = 0
⇒ (4x − 15)(x − 25) = 0 ⇒ (4x − 15) = 0 or (x − 25) = 0
⇒ $x=25orx=\frac{15}{4}$
⇒ x = 25 $\left[\mathrm{Since}\mathrm{x}=\frac{15}{4}\Rightarrow \left(\mathrm{x}-10\right)<0\right]$
Hence, the time taken by the smaller tap to fill the tank = 25 hours.
And the time taken by the larger tap to fill the tank = (25 − 10) hours = 15 hours.