Question

Solve for \(x\) in the following proportions. Carry division two decimal places as necessary. \(x: 9=5: 10\)

Short answer

\( x = 4.5 \)

Step-by-step solution

Understand the Proportion

A proportion is an equation that states that two ratios are equal. We are given \( \frac{x}{9} = \frac{5}{10} \). Our goal is to find the value of \( x \).

Cross-Multiply to Clear Fractions

To solve the proportion, we use cross-multiplication. Multiply the numerator of the first ratio by the denominator of the second ratio and the denominator of the first ratio by the numerator of the second ratio:\[ x \times 10 = 9 \times 5 \]

Simplify the Equation

Calculate the products from the cross-multiplication:\[ 10x = 45 \]

Isolate \(x\)

Divide both sides of the equation by 10 to solve for \( x \):\[ x = \frac{45}{10} = 4.5 \]

Check Your Answer

Plug \( x = 4.5 \) back into the original proportion to verify:\[ \frac{4.5}{9} = \frac{5}{10} \] which simplifies to \( 0.5 = 0.5 \), confirming our solution is correct.

Key concepts

Cross-Multiplication

Cross-multiplication is a crucial method for solving proportions. When you have two ratios set equal, such as \( \frac{x}{9} = \frac{5}{10} \), cross-multiplication helps us eliminate the fractions and work with whole numbers. The process involves multiplying the numerator (top number) of one ratio by the denominator (bottom number) of the other ratio. In our example, we perform the following:

• Multiply \( x \) (numerator of the first ratio) by 10 (denominator of the second ratio).
• Multiply 9 (denominator of the first ratio) by 5 (numerator of the second ratio).

This gives us the equation \( 10x = 45 \). The beauty of cross-multiplication is its simplicity; it transforms the problem into a straightforward equation that can be easily solved for the unknown value.

Ratio Simplification

Ratio simplification is an important step when verifying solutions in proportion problems. It ensures the ratios on both sides of the equation are equivalent when reduced to their simplest forms. Let's look at the solution \( \frac{4.5}{9} \) and \( \frac{5}{10} \).

• Both these fractions reduce to the ratio \( 0.5 \) when divided out.

To simplify \( \frac{4.5}{9} \), we divide the numerator and the denominator by their greatest common divisor if necessary. Here, dividing 4.5 by 9 directly gives us 0.5. Similarly, divide 5 by 10 to also get 0.5. When the simplified values on each side of a proportion are equal, it confirms that our solution is correct. This provides a useful check to ensure all calculations have been done properly.

Decimal Division

Decimal division often appears at the final stage of solving a proportion. After we use cross-multiplication, we frequently arrive at an equation that requires dividing a decimal. In this case, we had to solve \( 10x = 45 \).

• To isolate \( x \), we divide 45 by 10.

This gives us \( x = 4.5 \). Two important tips when dealing with decimal division:

• Remember to move the decimal point carefully, especially if you are dividing by a number like 10, 100, etc., where you can simply move the decimal point one or two places to the left.
• Round correctly if the problem asks for a specific number of decimal places; here, 4.5 is exact, so no rounding is needed.

Understanding decimal division helps to accurately determine the value of the unknown in proportion equations, ensuring the final solution is precise and correct.