Question

Solve for \(x\) in the following proportions. Carry division two decimal places as necessary. \(20: 40=x: 15\)

Short answer

x = 7.5

Step-by-step solution

Set Up the Proportion Equation

Start by writing the given proportion as an equation. The problem gives us \( \frac{20}{40} = \frac{x}{15} \).

Cross Multiply

Cross multiply to eliminate the fractions. This means you multiply 20 by 15 and x by 40, setting the two products equal to each other: \( 20 \times 15 = 40 \times x \).

Simplify the Equation

Calculate the product on the left side of the equation: \( 20 \times 15 = 300 \). Now the equation is \( 300 = 40x \).

Isolate x

Solve for \( x \) by dividing both sides of the equation by 40: \( x = \frac{300}{40} \).

Perform the Division

Complete the division to find the value of \( x \). Divide 300 by 40 to get \( x = 7.5 \).

Key concepts

Understanding Cross Multiplication

Cross multiplication is a method used to solve equations involving two fractions set equal to each other, called proportions. It helps eliminate the fractions and makes solving these equations easier. When a proportion is given, such as \( \frac{a}{b} = \frac{c}{d} \), cross multiplication involves multiplying the numerator of one fraction by the denominator of the other fraction, across the equal sign. Here's how you do it:
- Multiply the numerator of the first fraction by the denominator of the second fraction: \( a \times d \).
- Multiply the numerator of the second fraction by the denominator of the first fraction: \( b \times c \).
- You then set these two products equal to each other. So, \( a \times d = b \times c \).
This technique is efficient because it removes the fractions and provides a straightforward equation that can be solved using standard algebraic methods.

Steps for Equation Solving

Once you've cross-multiplied, you're left with an equation that is often simpler to solve. Here's a step-by-step approach to solving it:
- After cross multiplying, you'll have an equation like: \( m = n \), where \( m \) and \( n \) are the products from cross multiplying.
- The goal is to isolate the variable to one side of the equation. This often involves algebraic manipulations such as addition, subtraction, multiplication, or division.
For example, if you end with \( 40x = 300 \):
- Divide both sides by the coefficient of \( x \) (which is 40 in this case) to solve for \( x \).
So, \( x = \frac{300}{40} \).
Through this process, equation solving becomes a method of balancing the equation, ensuring both sides remain equal while simplifying to find the value of the unknown variable.

Simplifying Ratios

Simplifying ratios means reducing them to their simplest form, making them easier to interpret and work with. A ratio compares two quantities and is often expressed in terms such as 20:40.
- First, find the greatest common divisor (GCD) for both numbers in the ratio.
- Divide both terms of the ratio by the GCD to simplify.
In our example, 20 and 40 share a GCD of 20. By dividing both numbers by 20, the ratio 20:40 simplifies to 1:2.
Simplifying ratios helps to see the direct relationship between the numbers and can make other mathematical operations, like solving equations, more straightforward. This is especially helpful in proportion problems, where understanding the simplest form of a ratio might clarify the balance needed between different quantities.