The sum of two numbers is 184. If one third of the one exceeds one seventh of another by ‘8’ The smaller number is

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Detailed Solution

The image shows a problem involving two numbers with a total sum of 184. Here is the detailed solution and analysis of the steps presented:

Problem Breakdown:

Given:

The sum of two numbers = 184.
Let one of the numbers be $x$ $x$ .
The expression $x \times \frac{1}{3} - \frac{(184 - x)}{7} = 8$ $x$ $\times$ $31$ $$ $-$ $7$ $($ $184$ $-$ $x$ $)$ $$ $=$ $8$ is given to find $x$ $x$ .

Step 1: Write the Equation

x \times \frac{1}{3} - \frac{(184 - x)}{7} = 8

$x$ $\times$ $31$ $$ $-$ $7$ $($ $184$ $-$ $x$ $)$ $$ $=$ $8$

Simplify $x \times \frac{1}{3}$ $x$ $\times$ $31$ $$ :

\frac{x}{3} - \frac{184 - x}{7} = 8

$3$ $x$ $$ $-$ $7184$ $-$ $x$ $$ $=$ $8$

Step 2: Remove the Fractions

To eliminate the fractions, multiply the entire equation by the Least Common Denominator (LCD) of 3 and 7, which is 21:

21 \times \left(\frac{x}{3} - \frac{184 - x}{7}\right) = 21 \times 8

$21$ $\times$ $($ $3$ $x$ $$ $-$ $7184$ $-$ $x$ $$ $)$ $=$ $21$ $\times$ $8$

Simplify each term:

7x - 3(184 - x) = 168

$7$ $x$ $-$ $3$ $($ $184$ $-$ $x$ $)$ $=$ $168$

Step 3: Expand and Simplify

Expand $3(184 - x)$ $3$ $($ $184$ $-$ $x$ $)$ :

7x - 552 + 3x = 168

$7$ $x$ $-$ $552$ $+$ $3$ $x$ $=$ $168$

Combine like terms:

10x - 552 = 168

$10$ $x$ $-$ $552$ $=$ $168$

Step 4: Solve for $x$ $x$

Add 552 to both sides:

10x = 168 + 552

$10$ $x$ $=$ $168$ $+$ $552$

Simplify:

10x = 720

$10$ $x$ $=$ $720$

Divide by 10:

x = \frac{720}{10} = 72

$x$ $=$ $10720$ $$ $=$ $72$

x = 72

$x$ $=$ $72$

If $x = 72$ $x$ $=$ $72$ , the other number is $184 - 72 = 112$ $184$ $-$ $72$ $=$ $112$ .
Substitute $x = 72$ $x$ $=$ $72$ into the original equation: $\frac{72}{3} - \frac{112}{7} = 8$ $372$ $$ $-$ $7112$ $$ $=$ $8$ Simplify: $24 - 16 = 8$ $24$ $-$ $16$ $=$ $8$ The equation is satisfied, confirming that $x = 72$ $x$ $=$ $72$ is correct.