Question

The circumference of the front wheel of a cart is $30$ ft. long and that of the back wheel is $36$ ft. What is the distance travelled by the cart when the front wheel has done $5$ more revolutions than the real wheel.
A) 20
B) 25
C) 750
D) 900

Answer 1

Hint:The distance travelled by the front wheel will be equal to the distance travelled by the rear wheel.

Complete step-by-step solution:
The front wheel is smaller in diameter than the rear wheel.
Therefore, the revolutions made by the front wheel will be more than the rear wheel, for the given speed of the bicycle.
Let the number of revolutions made by the rear wheel be $x$
Then, the number of revolutions made by the front wheel is $5$ more than the front wheel is given by,$x + 5$
The distance travelled by the front wheel will be equal to the distance travelled by the rear wheel despite the difference in their circumference.
The distance travelled by the rear wheel is
${d_1} = $ Number of revolutions made by rear wheel x Circumference of the rear wheel.
$
 \Rightarrow {d_1} = x \times 36 \\
 \Rightarrow {d_1} = 36x \cdots \left( 1 \right) \\
 $
${d_2} = $ Number of revolutions made by front wheel x Circumference of the front wheel.
$
\Rightarrow {d_2} = \left( {x + 5} \right) \times 30 \\
 \Rightarrow {d_2} = 30x + 150 \cdots \left( 2 \right) \\
 $
According to the condition, both equation (1) and (2) are equal.
$36x = 30x + 150$
Solving for will give the number of revolutions made by the rear wheel.
$
  \Rightarrow 36x - 30x = 150 \\
\Rightarrow 6x = 150 \\
\Rightarrow x = 25 \\
 $
The number of revolutions made by the rear wheel is $25$
The number of revolutions made by the front wheel is
$
\Rightarrow x + 25 = 5 + 25 \\
   = 30 \\
 $
Therefore, the distance travelled by the cart in terms of rear wheel is ,
$D = 36x \cdots \left( 3 \right)$
Substituting the value of $x = 25$ in equation (3),
$
 \Rightarrow D = 36 \times 25 \\
\Rightarrow D = 900 \\
 $
The distance travelled by the cart is $900$ ft.
Hence, the correct option is (D).

Note:The important step is to realize that the distance travelled by the front wheel is equal to the distance travelled by the rear wheel.
The distance travelled can be calculated in terms of front wheel is given by,
$D = 30x + 150 \cdots \left( i \right)$
Substitute the value of $x = 25$ in equation (i), we get
$
\Rightarrow D = 30 \times 25 + 150 \\
\Rightarrow D = 900 \\
 $
The distance travelled by the cart is $900$ ft.