Question

10. A fruits seller has fruits of which 1 /2 potion guava, 1/3 portion is orange and 50 mangoes.


(a) Find the total portion of guava and orange when number of total fruits being x

(b) find the total number of fruits

(c) The difference between two positive integers is 50 and there ratio is 1 :4. find these two numbers. ​

Answer 1

Given,

Number of mangoes $\[ = 50\]$

Number of guava portion$\[ = \frac{1}{2}\]$ portion

Number of orange $\[ = \frac{1}{3}\]$ portion

Solution,

Consider the total number of fruits are x.

$\[\begin{array}{l}\frac{1}{2}x + \frac{1}{3}x + 50 = x\\ \Rightarrow - \frac{1}{2}x + \frac{1}{3}x + 50 = 0\\ \Rightarrow - \frac{1}{6}x = - 50\\ \Rightarrow x = 300\end{array}\]$

(a) Find the total portion of guava and orange when number of total fruits being x.

$\[\begin{array}{l} = \frac{1}{2}x + \frac{1}{3}x\\ = \frac{5}{6}x\end{array}\]$

(b) find the total number of fruits.

$\[\begin{array}{l}\frac{1}{2}x + \frac{1}{3}x + 50 = x\\ \Rightarrow - \frac{1}{2}x + \frac{1}{3}x + 50 = 0\\ \Rightarrow - \frac{1}{6}x = - 50\\ \Rightarrow x = 300\end{array}\]$

Hence the total number of fruits are 300.

(c) The difference between two positive integers is 50 and there ratio is 1 :4. find these two numbers. ​

Consider the positive integers are x and y.

$\[x - y = 50\]$--------(1)

$\[\frac{x}{y} = \frac{1}{4}\]$

$\[ \Rightarrow y = 4x\]$--------(2)

Put the value of y from equation 2 to equation 1,

$\[\begin{array}{l}x - 4x = 50\\ \Rightarrow - 3x = 50\\ \Rightarrow x = - \frac{{50}}{3}\end{array}\]$

Next step,

$\[\begin{array}{l}y = 4x\\ \Rightarrow y = 4 \times \left( { - \frac{{50}}{3}} \right)\\ \Rightarrow y = - \frac{{200}}{3}\end{array}\]$

Hence the integers are$\[ - \frac{{50}}{3}\]$ and $\[ - \frac{{200}}{3}\]$.