Question

Translate to an equation and solve. \(120 \%\) of what number is \(65.52 ?\)

Short answer

Question: Find the number, which when multiplied by 120%, gives the result 65.52. Answer: 54.6

Step-by-step solution

Write the equation

Let the unknown number be \(x\). To find 120% of a number, we multiply it by \(\frac{120}{100}\). So, we can write the equation as: $$ \frac{120}{100} \cdot x = 65.52 $$

Solve for x

To solve for \(x\), we need to isolate it. We do this by multiplying both sides of the equation by the reciprocal of \(\frac{120}{100}\). The reciprocal of \(\frac{120}{100}\) is \(\frac{100}{120}\). Hence, the equation becomes: $$ x = 65.52 \cdot \frac{100}{120} $$ Now we can calculate the value of \(x\): $$ x = 65.52 \cdot \frac{100}{120} = 65.52 \cdot \frac{5}{6} = 54.6 $$ So, 120% of the number 54.6 is 65.52.

Key concepts

Percentage Calculation

Understanding percentage calculation is a fundamental skill, useful in a variety of real-world scenarios such as finance, statistics, and even in everyday shopping discounts. When we refer to a percentage, we're talking about a portion out of 100. To calculate the percentage of a number, we convert the percentage to its decimal form by dividing it by 100, then multiply by the number in question.

For example, to calculate 120% of a number, you would convert 120% to a decimal (which is 1.2), and then multiply that by the number. If we're given that 120% of a number is 65.52, we set up the equation \( 1.2 \times x = 65.52 \), where \(x\) represents the unknown number.

Algebraic Equations

Algebraic equations are like puzzles. They involve variables, numbers, and operations that together represent a statement of equality. Solving an algebraic equation means finding the value of the variable that makes the equation true.

In the context of our example, the equation \( 1.2 \times x = 65.52 \) involves finding the value of \(x\) that satisfies the condition. The beauty of algebra is that it gives us a systematic approach to solve for the unknown by performing the same operations on both sides of the equation. This balance is the core of algebraic manipulation and helps ensure the two sides of the equation remain equal.

Isolating Variables

Isolating variables is a critical step in solving algebraic equations; it's essentially getting the unknown all by itself on one side of the equation. This process often involves reversing operations that are being applied to the variable.

In our example, we start with \(1.2 \times x = 65.52\), and we want to get \(x\) by itself. To do that, we must eliminate the 1.2 that is being multiplied by \(x\).

Reciprocal to the Rescue

Multiplying by the reciprocal of 1.2 (which is \(\frac{5}{6}\) because \(\frac{1}{1.2} = \frac{5}{6}\)) will cancel out the 1.2, leaving us with \(x\) on one side. This step simplifies the equation dramatically, converting a seemingly complicated problem into a much simpler one.

Reciprocal of a Number

The reciprocal of a number is what you multiply the number by to get 1. It's like a 'mathematical flip'. If you have a fraction, you flip its numerator and denominator to get its reciprocal. For a whole number or an integer, its reciprocal is 1 divided by the number.

In our solved algebraic equation, \(\frac{120}{100} \cdot x = 65.52\), we used the reciprocal of \(\frac{120}{100}\), which is \(\frac{100}{120}\) or simplified as \(\frac{5}{6}\). Multiplying both sides of the equation by \(\frac{5}{6}\) allowed us to cancel out \(\frac{120}{100}\) and isolate \(x\). Understanding reciprocals is essential to solving equations and simplifying expressions effectively.