Question

Solve for \(x\) in the following proportions. Carry division two decimal places as necessary. \(\frac{2.2}{x}=\frac{8.8}{5}\)

Short answer

The value of \(x\) is 1.25.

Step-by-step solution

Understand the Proportion

We have the proportion \(\frac{2.2}{x} = \frac{8.8}{5}\). This means two ratios are equal. To solve for \(x\), we need to manipulate this equation to isolate \(x\).

Cross-Multiply

Cross-multiply the terms to eliminate the fractions. This gives us: \(2.2 \times 5 = 8.8 \times x\).

Simplify the Equation

After cross-multiplying, we get the equation: \(11 = 8.8 \times x\). This simplifies the right side of the equation, allowing us to solve for \(x\).

Solve for x

We now need to isolate \(x\) by dividing both sides of the equation by 8.8: \(x = \frac{11}{8.8}\).

Perform the Division

Calculate \(\frac{11}{8.8}\) by performing the division. Carry the division to two decimal places: \(x = 1.25\).

Key concepts

Cross-Multiplication

Cross-multiplication is a fundamental technique in mathematics used to solve equations that involve proportions. When you have a proportion, like \(\frac{a}{b} = \frac{c}{d}\), cross-multiplication helps you eliminate the fractions by transforming the equation into a simpler form.
Here’s how it works:

• Multiply the numerator of the first ratio with the denominator of the second ratio.
• And the numerator of the second ratio with the denominator of the first ratio.

This gives the new equation: \(a \times d = c \times b\). By setting these cross-products equal, you effectively remove the fractions, making the equation easier to solve.
In our exercise where \(\frac{2.2}{x} = \frac{8.8}{5}\), cross-multiplying gives us the equation \(2.2 \times 5 = 8.8 \times x\), simplifying the process of finding \(x\).

Solving for x

Solving for \(x\) is about isolating \(x\) on one side of the equation, which allows you to determine its value. Once you've cross-multiplied the terms in the proportion, you get a simple equation that you need to solve for \(x\).
In the example, after cross-multiplication, we obtained \(11 = 8.8 \times x\). The goal is to have \(x\) by itself on one side of the equation. To do this, divide both sides of the equation by 8.8:\[x = \frac{11}{8.8}\]This step effectively "solves for \(x\)" because it isolates \(x\) and presents its value in terms of simple arithmetic operations.
Remember, the key to solving for \(x\) is:

• Perform the same operation on both sides of the equation.
• Keep \(x\) isolated until you find its value.

Ratio and Proportion

Ratio and proportion are mathematical concepts that express the relationship between numbers and quantities. A ratio is a comparison of two numbers, while a proportion states that two ratios are equal.
Consider a ratio \(\frac{a}{b}\), which compares \(a\) to \(b\). When you set two ratios equal to each other like \(\frac{a}{b} = \frac{c}{d}\), this is a proportion. This means the relationship between \(a\) and \(b\) is equivalent to the relationship between \(c\) and \(d\).
In our example, \(\frac{2.2}{x} = \frac{8.8}{5}\), the proportion tells us that the ratio of 2.2 to \(x\) is the same as 8.8 to 5. Proportions are solved using cross-multiplication, making them a powerful tool for solving equations involving unknowns.

Division with decimals

Division with decimals can sometimes be tricky, but with some practice, it becomes manageable. In solving proportions, your final step might involve dividing to find the unknown value.
When you have a fraction with a decimal as the denominator, such as \(x = \frac{11}{8.8}\), you perform long division to find \(x\). Here, it's important to round to the specified decimal places, usually two, to ensure precision.
Steps to divide with decimals:

• Move the decimal point of the divisor to the right till it's a whole number.
• Move the dividend’s decimal point the same number of spaces to the right.
• Perform the division as usual.

In our exercise, dividing 11 by 8.8 gives \(x = 1.25\), showing that even with decimals, division can be straightforward once you adjust the decimal appropriately.