Question

In a picnic, there are boys and girls. Fifteen girls leave, then the boys and girls are left in the ratio of\[2:1\]. Later 45 boys leave and the ratio changes to\[1:5\]. The numbers of girls, in the beginning, was 8p then p is

Answer 1

Hint: Ratio is a quantitative relation between two amounts, which shows the number of times one value contains with the other. It is a way to compare two quantities by using the division method, where a mile per hour is a comparison of miles and hours. It is a mathematical expression written in the form a: b, where a and b be any integers.

Complete step-by-step answer:
Let the number of the girls, in the beginning, be 8p and the number of the boys q, whose ratio is given as \[\dfrac{q}{{8p}}\]
Since fifteen numbers of girls leave the picnic, the equation is given as
 \[\dfrac{2}{1} = \dfrac{q}{{8p - 15}}\]
The above equation can be re-written as:
\[
  \dfrac{2}{1} = \dfrac{q}{{8p - 15}} \\
  8p - 15 = \dfrac{q}{2} - - - (i) \\
 \]
Now 45 numbers of boys leave the picnic hence the ration changes from \[2:1\]to\[1:5\], therefore the ratio becomes
\[\dfrac{1}{5} = \dfrac{{q - 45}}{{8p - 15}}\]
By cross multiplying, we get
\[
  \dfrac{1}{5} = \dfrac{{q - 45}}{{8p - 15}} \\
  8p - 15 = 5\left( {q - 45} \right) \\
  8p - 15 = 5q - 225 - - - (ii) \\
 \]
Now equate equation (i) and (ii) in terms of\[8p - 15\], we get
\[
  \dfrac{q}{2} = 5q - 225 \\
  q = 10q - 450 \\
  10q - q = 450 \\
  9q = 450 \\
  q = \dfrac{{450}}{9} \\
   = 50 \\
 \]
Now substitute the value of q in equation (i) to find the value of p,
\[
  8p - 15 = \dfrac{q}{2} \\
  8p - 15 = \dfrac{{50}}{2} \\
  8p = 25 + 15 \\
  p = \dfrac{{40}}{8} \\
   = 5 \\
 \]
Hence the number of girls in the beginning\[p = 5\]

Note: Students must note that the ratio is the relation between two quantities and if anyone of these quantities changes then the whole ratio changes.