Question

Solve for \(x\) in the following proportions. Carry division two decimal places as necessary. \(\frac{5}{7}=\frac{x}{35}\)

Short answer

The value of \(x\) is 25.

Step-by-step solution

Cross Multiply

To solve the proportion \( \frac{5}{7} = \frac{x}{35} \), begin by cross-multiplying to eliminate the fractions. This results in two products: \(5 \times 35\) and \(7 \times x\). The equation becomes \(5 \times 35 = 7x\).

Simplify Products

Calculate each product. Start with \(5 \times 35\), which equals 175. So the equation now reads \(175 = 7x\).

Isolate x

To solve for \(x\), divide both sides of the equation by 7. This gives \(x = \frac{175}{7}\).

Perform Division

Divide 175 by 7 to find the value of \(x\). \(175 \div 7 = 25\).

Conclude Solution

After dividing, you determine that \(x = 25\).

Key concepts

Cross Multiplication

Cross multiplication is a nifty tool used when solving equations involving proportions. A proportion is an equation where two ratios are set equal to each other, like this: \( \frac{a}{b} = \frac{c}{d} \). The goal of cross multiplication is to eliminate the fractions, making the equation easier to solve.
To cross-multiply, you multiply the diagonal terms across the equal sign. So, with our proportion, you multiply \( a \) by \( d \) and \( b \) by \( c \), giving you the equation \( a \times d = b \times c \).
This method effectively transforms a complex fraction problem into a more straightforward multiplication problem. It's especially useful in situations where both sides of the equation can be simplified after clearing the fractions.
Typically, you will follow these steps when cross-multiplying:

• Identify the two cross-products in the proportion.
• Write the equation of the products.
• Solve the resulting equation.

Cross multiplication is a very powerful step in solving proportions swiftly and accurately.

Proportion

Understanding the concept of a proportion is crucial in many mathematical contexts. A proportion compares two ratios and suggests that they are equivalent. This is often expressed as two fractions being equal, like \( \frac{a}{b} = \frac{c}{d} \).
In real-world terms, proportions can help us understand and solve problems relating to scaling, conversion, and even financial math. Let's break it down further:

• The first term of the ratio (\( a \)) is known as the numerator, while the second term (\( b \)) is the denominator in the first fraction.
• Similarly, the same naming convention applies to the second fraction (\( \frac{c}{d} \)).
• A true proportion means multiplying cross-terms will yield equal products, i.e., \( a \times d = b \times c \).

This equality forms the foundation of a proportion. By applying cross multiplication, the fraction disappears, reducing it to a simple equation that can be solved easily for an unknown variable.

Solving Equations

The art of solving equations is a fundamental skill in mathematics. Once you've performed cross multiplication to clear fractions from a proportion, you're often left with an algebraic equation to solve.
The equation represents a balance; whatever you do to one side of the equation, you must do to the other to maintain equality.
Here's a step-by-step thought process for solving equations derived from proportions:

• First, perform any necessary multiplication or division to simplify the equation further.
• Next, isolate the variable (often represented as \( x \)) on one side of the equation. You do this by performing inverse operations, such as dividing if the variable is multiplied by a number.
• Finally, simplify the solution to find the value of the unknown variable.

In the specific example given, after cross multiplication and simplification, we solved \( 7x = 175 \) by dividing both sides by 7, which resulted in \( x = 25 \). This balance is key in finding the correct solution.