Sure! Let's solve the problem step-by-step.
1. Define the variables:
- Let the other number be [tex]\( x \)[/tex].
- The smaller number is [tex]\( \frac{2}{3} \)[/tex] of the other number, hence it is [tex]\( \frac{2}{3} x \)[/tex].
2. Set up the equation based on the given information:
- The sum of the two numbers is given as 30.
- Therefore, the equation is:
[tex]\[
x + \frac{2}{3} x = 30
\][/tex]
3. Combine like terms:
- To simplify this, we can combine the terms involving [tex]\( x \)[/tex]:
[tex]\[
x + \frac{2}{3} x = 30
\][/tex]
[tex]\[
\left(1 + \frac{2}{3}\right) x = 30
\][/tex]
4. Simplify the coefficient:
- We need to add the fractions:
[tex]\[
1 + \frac{2}{3} = \frac{3}{3} + \frac{2}{3} = \frac{5}{3}
\][/tex]
- Therefore, the equation becomes:
[tex]\[
\frac{5}{3} x = 30
\][/tex]
5. Solve for [tex]\( x \)[/tex]:
- To isolate [tex]\( x \)[/tex], multiply both sides by the reciprocal of [tex]\( \frac{5}{3} \)[/tex], which is [tex]\( \frac{3}{5} \)[/tex]:
[tex]\[
x = 30 \times \frac{3}{5}
\][/tex]
- Perform the multiplication:
[tex]\[
x = \frac{90}{5} = 18
\][/tex]
6. Find the smaller number:
- The smaller number is [tex]\( \frac{2}{3} \)[/tex] of the other number [tex]\( x \)[/tex]:
[tex]\[
\text{Smaller number} = \frac{2}{3} \times 18 = 12
\][/tex]
So, the smaller number is [tex]\( 12 \)[/tex].
