Question

Solve for \(x\) in the following proportions. Carry division two decimal places as necessary. \(20: 40=x: 10\)

Short answer

The value of \(x\) is 5.00.

Step-by-step solution

Identify the Proportion

The problem gives us a proportion in the form of a ratio: \(20:40 = x:10\). This can be rewritten in fraction form as \(\frac{20}{40} = \frac{x}{10}\).

Cross-Multiply to Solve for x

To solve for \(x\), cross-multiply the fractions. This means multiply the numerator of the first fraction by the denominator of the second fraction and the denominator of the first fraction by the numerator of the second fraction. This gives us: \[20 \times 10 = 40 \times x\] which simplifies to \[200 = 40x\].

Solve for x

Now, isolate \(x\) by dividing both sides of the equation by 40. This gives us: \[x = \frac{200}{40}\]. Calculate the division to find \(x\).

Calculate the Division

Perform the division: \[x = \frac{200}{40} = 5\]. With division carried out to two decimal places, \(x = 5.00\).

Key concepts

Cross Multiplication

Cross multiplication is an essential technique in solving proportion problems. It simplifies the process of comparing two fractions or ratios. When you cross-multiply, you take the numerator (the top number) of one fraction and multiply it by the denominator (the bottom number) of the other fraction. This eliminates the fractions and turns the problem into a simpler equation.

For the proportion \( \frac{20}{40} = \frac{x}{10} \), cross-multiplication involves:

• Multiplying 20 (numerator of the first fraction) by 10 (denominator of the second fraction).
• Multiplying 40 (denominator of the first fraction) by \(x\) (numerator of the second fraction).

These operations lead to the equation: \[ 20 \times 10 = 40 \times x \] Which simplifies to: \[ 200 = 40x \]This equation without fractions is much easier to solve, allowing you to find the value of \(x\) by further simplifying the equation.

Ratio and Proportion

Ratios and proportions are mathematical ways to show the relationship between two numbers or quantities. A ratio is a comparison between two numbers, often expressed with a colon, like 20:40. A proportion, meanwhile, is an equation that states that two ratios are equal.

In the problem \(20:40 = x:10\), we are given a proportion, and the task is to find the unknown value \(x\) that will make these two ratios equal.

• The ratio 20:40 can simplify to 1:2 (by dividing both sides by 20).
• Similarly, the ratio \(x:10\) must also equal 1:2 to maintain the proportion.

Understanding ratios and proportions helps to set up and solve equations accurately. You recognize that preserving the balance or equality between the ratios is key to solving such problems.

Solving for Variables

Solving for variables in mathematical equations involves isolating the variable on one side of the equation to find its value. This is what you aim to do once the proportion equation is set.

Using the equation derived from cross-multiplication, \( 200 = 40x \), you can solve for \(x\) by moving all other numbers away from \(x\). Here’s how:

• Divide both sides of the equation by the coefficient of \(x\), which is 40.

So you have: \[ x = \frac{200}{40} \]This calculation results in: \[ x = 5 \] Performing division and simplifying gives you the precise value of the variable. It shows how proportional relationships and simple arithmetic can lead you to the solution. As a result, the value of \(x\) is given in the proportion \(x:10\), which satisfies the original condition that \(20:40 = x:10\).