# 03 - Excess of the Sum of Parts over the Difference of Parts of a Number

**Problem**

The excess of the sum of the half and the third parts over the difference of the fourth and the fifth parts of a number is 188. What is the value of the number?

**Solution**

Let x = the number

x/2 = the half part of x

x/3 = the third part of x

x/4 = the fourth part of x

x/5 = the fifth part of x

x/2 = the half part of x

x/3 = the third part of x

x/4 = the fourth part of x

x/5 = the fifth part of x

S_{1} = sum of the half and third parts

$S_1 = \dfrac{x}{2} + \dfrac{x}{3}$

S_{2} = difference of fourth and fifth parts

$S_2 = \dfrac{x}{4} - \dfrac{x}{5}$

The excess of S_{1} over S_{2} = 188

$S_1 - S_2 = 188$

$\left( \dfrac{x}{2} + \dfrac{x}{3} \right) - \left( \dfrac{x}{4} - \dfrac{x}{5} \right) = 188$

$\frac{47}{60}x = 188$

$x = 240$ *answer*

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