Question

Translate to an equation and solve. \(65 \%\) of what number is \(52 ?\)

Short answer

Answer: 80

Step-by-step solution

Translate the problem into an equation

Let the unknown number be represented as "x". Then we can express the problem as: \(65 \%\) of "x" = \(52\) Since percentages are expressed as a fraction over 100, we can write the equation as: \(\frac{65}{100}\times x = 52\)

Solve the equation

To solve the equation for x, we need to isolate x by dividing both sides of the equation by \(\frac{65}{100}\): \(x = \frac{52}{\frac{65}{100}}\) Now, we can simplify the fraction by multiplying both the numerator and denominator by 100: \(x = \frac{52 \times 100}{65}\) Finally, we can divide 52 by 65: \(x = \frac{5200}{65} = 80\) So, 65% of the number 80 is equal to 52.

Key concepts

Translating Word Problems

Translating word problems into mathematical equations is a critical skill in understanding and solving problems you may encounter in mathematics. It involves recognizing keywords and phrases that indicate mathematical operations and relationships. In the case of percentage word problems, keywords such as 'of', 'is', 'percent', 'what number', indicate a multiplication operation where 'of' means multiply, and 'percent' means 'per hundred'.

For our specific problem, we translate '65% of what number is 52?' to an equation. We identify '65%' as the percentage, 'of' as multiplication, 'what number' as our unknown variable, often denoted by 'x', and 'is' indicating equality. The translation, therefore, is \(65\% \times x = 52\). By converting the percentage to a decimal or fraction, we turn the word problem into an equation we can solve.

Solving Equations

Solving equations is about finding the value of the unknown that makes the equation true. In percentage problems, once we've translated the problem into an equation, we solve for the unknown variable. Starting with an equation like \(\frac{65}{100} \times x = 52\), we isolate the variable.To do this, we perform inverse operations to get 'x' by itself. Here, we start by dividing both sides of the equation by \(\frac{65}{100}\), essentially reversing the multiplication. Doing so gives us \(x = \frac{52}{\frac{65}{100}}\), which can be simplified further to \(x = \frac{5200}{65}\). By simplifying the fraction, we find the value of 'x'. It’s essential to perform the same operations on both sides of the equation to maintain balance and eventually find the correct solution for 'x'.

Percentage Calculations

Percentage calculations often seem daunting, but they can be made simpler with practice and by understanding the basic concept that a percentage is a way to express a number as a fraction of 100. To work with percentages in equations, you can convert them into decimals or fractions. For example, 65% can be written as \(0.65\) or \(\frac{65}{100}\).In the equation \(x = \frac{52}{\frac{65}{100}}\), we see percentage calculations in action. When simplifying it, we are converting the percentage to its fraction form, making it easier to solve for 'x'. Multiplying the numerator and denominator by 100 allows us to clear the fractional denominator and find that \(52\) is \(65%\) of \(80\). Remembering that percentages are parts of a whole can help demystify these calculations, making it easier to understand and solve the problems you’re faced with.