Question

Translate to an equation and solve. What percent is 169 of \(180 ?\)

Short answer

Answer: Approximately 93.89%

Step-by-step solution

Set up the proportion.

We are asked to find what percent 169 is of 180, so let x be the unknown percentage. We write the proportion as follows: \(\frac{169}{180} = \frac{x}{100}\)

Solve for x.

In order to solve for x, we can cross multiply: \(169 * 100 = 180 * x\) Now, divide both sides by 180: \(x = \frac{169 * 100}{180}\)

Calculate the result.

Now we can calculate the value of x: \(x = \frac{16900}{180} \approx 93.89\)

Interpret the result.

The value of x is approximately 93.89. This means that 169 is approximately 93.89% of 180.

Key concepts

Solving Equations

Solving equations is a fundamental skill in mathematics that helps us find unknown values by establishing relationships between different quantities. When faced with an equation, the primary goal is to isolate the variable, which is often represented by a letter, such as \( x \), on one side of the equation. This process typically involves performing operations that will simplify the equation and bring us closer to the solution.

In the provided exercise, we initially set up an equation as a proportion with:

• 169 (the part we know) over 180 (the part we compare to), equal to the unknown percentage \( x \) over 100.

This equation is represented as:
\[ \frac{169}{180} = \frac{x}{100} \]
The solution involves manipulating this equation to determine the value of \( x \). By understanding equations and knowing how to manipulate them, you can solve a wide range of mathematical problems.

Proportion

A proportion is an equation that states two ratios are equal. Proportions are very useful when comparing quantities and finding unknown numbers based on the relationships between known quantities. They can be identified by the format \( \frac{a}{b} = \frac{c}{d} \), where two fractions are set equal to one another.

The exercise demonstrates a proportion by setting the ratio of 169 to 180 equal to the ratio of \( x \) to 100. This specific proportion is designed to find out what percentage 169 is of 180. When you see a proportion, you can solve it using techniques such as cross multiplication. This method allows us to eliminate the fractions and easily solve for the unknown variable.

Cross Multiplication

Cross multiplication is a technique used to solve proportions. It involves multiplying across the diagonal of the equation to eliminate the fractions and simplify the calculation. This is especially helpful when you have equations in the form \( \frac{a}{b} = \frac{c}{d} \).

To apply cross multiplication, follow these steps:

• Multiply the numerator of the first ratio by the denominator of the second ratio.
• Multiply the denominator of the first ratio by the numerator of the second ratio.

The result is an equation without fractions:
\[ a \times d = b \times c \]
In the given exercise, this method was used by multiplying 169 by 100 and 180 by \( x \), leading to the equation: \[ 169 \times 100 = 180 \times x \]
Once you have this form, solving for \( x \) becomes straightforward by dividing both sides by 180. Cross multiplication simplifies the solving process significantly, making it an essential technique in solving proportions.