Question

Translate to an equation and solve. \(35 \%\) of what number is \(77 ?\)

Short answer

Answer: The number is 220.

Step-by-step solution

Convert percentage to decimal

We need to convert \(35 \%\) to its decimal form. To do this, divide 35 by 100. \(35 \% = \frac{35}{100} = 0.35\)

Set up an equation

Let the number we are looking for be represented by \(x\). We know that \(35 \%\) of this number is \(77\), so we can represent this relationship with the following equation: \(0.35 \times x = 77\)

Solve the equation

To isolate \(x\), we need to divide both sides by \(0.35\): \(x = \frac{77}{0.35}\) Now, calculate the value of \(x\): \(x = 220\) The number we were looking for is \(220\).

Key concepts

Understanding Percentages

A percentage is a way to express a number as a part of a whole. Specifically, it is a fraction with a denominator of 100. For example, 35% means 35 out of every 100. This is a very common way to compare quantities and is particularly useful in everyday life.

Percentages help us understand proportions and ratios. They are used in setting discounts, calculating interest rates, and analyzing data in various fields.

• To find a percentage of a number, you multiply the number by the percentage in decimal form.
• Conversely, to find what percentage one number is of another, you divide the part by the whole and multiply by 100.

Understanding how to work with percentages creates a strong foundation for solving many practical problems.

Solving Equations

An equation is a mathematical statement that asserts the equality of two expressions. In the case of finding the number in our problem, we use a simple equation to isolate the variable. Here, the equation is a tool to express the relationship given in a word problem more clearly.

When solving equations, the goal is to find the value of the unknown variable. To do this, you will perform operations that simplify the equation, ultimately isolating the variable on one side.

• For example, if you start with an equation like \(0.35 \times x = 77\), you need to perform the reverse operation to isolate \(x\), which involves dividing both sides by 0.35.
• Solving equations is a fundamental skill in mathematics, allowing you to apply logical steps to reach a solution.

Practicing with different types of equations strengthens your problem-solving capabilities.

Converting Percentages to Decimals

To work effectively with percentages in mathematical equations, converting them to decimals is essential. This conversion simplifies calculations and helps when setting up and solving equations.

Converting a percentage to a decimal is straightforward:

• Start by dividing the percentage by 100. For example, 35% becomes \(\frac{35}{100} = 0.35\).
• This conversion is important because it allows you to use the percentage directly in multiplication and division within equations.

Remember, moving the decimal point two places to the left also transforms the percentage into a decimal. This method can be particularly handy in mental math.
Being comfortable with this conversion makes dealing with percentages in equations much easier, ensuring accurate results.