Problem Analysis:
The task involves solving proportions using the method of cross-multiplication. Proportions are equations that state two ratios are equal, and they can be solved by multiplying diagonally (cross-multiplying) to eliminate the fractions.
#### Example 1:
The given proportion is:
\[
\frac{3}{x} = \frac{6}{8}
\]
#### Solution for Example 1:
1.
Cross-Multiply:
Multiply the numerator of the first fraction by the denominator of the second fraction, and the numerator of the second fraction by the denominator of the first fraction:
\[
3 \cdot 8 = 6 \cdot x
\]
2.
Simplify:
Perform the multiplication:
\[
24 = 6x
\]
3.
Solve for \( x \):
Divide both sides of the equation by 6 to isolate \( x \):
\[
x = \frac{24}{6} = 4
\]
4.
Verify:
Substitute \( x = 4 \) back into the original proportion to check:
\[
\frac{3}{4} = \frac{6}{8}
\]
Simplify both sides:
\[
\frac{3}{4} = \frac{3}{4} \quad \text{(True)}
\]
Thus, the solution for Example 1 is:
\[
\boxed{x = 4}
\]
---
#### Example 2:
The given proportion is:
\[
\frac{5}{x + 2} = \frac{3}{x - 1}
\]
#### Solution for Example 2:
1.
Cross-Multiply:
Multiply the numerator of the first fraction by the denominator of the second fraction, and the numerator of the second fraction by the denominator of the first fraction:
\[
5 \cdot (x - 1) = 3 \cdot (x + 2)
\]
2.
Expand Both Sides:
Distribute the terms on both sides:
\[
5x - 5 = 3x + 6
\]
3.
Isolate \( x \):
Subtract \( 3x \) from both sides to get all \( x \)-terms on one side:
\[
5x - 3x - 5 = 6
\]
Simplify:
\[
2x - 5 = 6
\]
Add 5 to both sides to isolate the term with \( x \):
\[
2x = 11
\]
Divide both sides by 2:
\[
x = \frac{11}{2}
\]
4.
Verify:
Substitute \( x = \frac{11}{2} \) back into the original proportion to check:
\[
\frac{5}{\frac{11}{2} + 2} = \frac{3}{\frac{11}{2} - 1}
\]
Simplify the denominators:
\[
\frac{5}{\frac{11}{2} + \frac{4}{2}} = \frac{3}{\frac{11}{2} - \frac{2}{2}}
\]
\[
\frac{5}{\frac{15}{2}} = \frac{3}{\frac{9}{2}}
\]
Simplify the fractions:
\[
\frac{5 \cdot 2}{15} = \frac{3 \cdot 2}{9}
\]
\[
\frac{10}{15} = \frac{6}{9}
\]
Simplify further:
\[
\frac{2}{3} = \frac{2}{3} \quad \text{(True)}
\]
Thus, the solution for Example 2 is:
\[
\boxed{x = \frac{11}{2}}
\]
---
Final Answers:
-
Example 1: \( \boxed{x = 4} \)
-
Example 2: \( \boxed{x = \frac{11}{2}} \)
Parent Tip: Review the logic above to help your child master the concept of solving ratios worksheet.
