Question

Translate to an equation and solve. \(105 \%\) of what number is \(49.14 ?\)

Short answer

Answer: The number is approximately 46.8.

Step-by-step solution

Understand the Problem

We are asked to find a number whose 105% is equal to 49.14. Let's call this unknown number x. Our goal is to find the value of x.

Set up the equation

We know that 105% is equal to 1.05 (as a decimal). Therefore, 105% of x can be expressed as 1.05x. Now, we set up our equation: 1.05x = 49.14

Solve the equation for x

To solve for x, we need to get it by itself on one side of the equation. We do this by dividing both sides of the equation by 1.05: x = \frac{49.14}{1.05}

Calculate the value of x

Now, we can calculate the value of x: x = \frac{49.14}{1.05} \approx 46.8

State the Answer

Therefore, 105% of 46.8 is 49.14.

Key concepts

Percentage Calculation

Understanding percentage calculation is pivotal when solving problems involving proportions and ratios that relate to a whole. In essence, percentage represents a fraction of 100, making it a convenient way to express fractions and decimals. For instance, when we say 105%, we are actually referring to 105 out of 100 or, in decimal form, 1.05.

To convert from a percentage to a decimal, you divide by 100, thus dropping the percent sign and moving the decimal two places to the left. Similarly, to convert from a decimal to a percentage, you multiply by 100, which means shifting the decimal two places to the right and adding a percent sign. It's important for students to be comfortable with these conversions as they frequently appear in various mathematical contexts, from simple interest calculations to more complex statistical data analysis.

Equation Solving Steps

The process of equation solving typically follows a set of steps, which could vary slightly based on the complexity of the equation in question. Generally, it starts with

Understanding the Problem

which involves determining what the question is asking for and identifying the unknowns. Next is

Setting Up the Equation

where the relationship between known and unknown values is expressed mathematically.

Following this, the

Isolation of the Variable

occurs, whereby operations are performed to get the unknown by itself on one side of the equation. This often involves inverse operations, such as dividing where there is multiplication or adding where there is subtraction. The final steps are

Performing the Calculations

and

Verifying the Solution

to ensure it makes sense in relation to the problem. Students must grasp these steps to effectively solve equations and validate their answers.

Translating Word Problems to Equations

A common challenge students face is translating word problems into equations. This skill requires a systematic approach to break down the language of the problem into mathematical expressions. It begins with

Identifying Key Information

and determining what the question is asking you to find. Words like 'is', 'of', and 'what' can help pinpoint the relationships and operations needed. The next part is

Assigning Variables

to unknown quantities, commonly symbolized as 'x' or 'y'.

After variables are assigned, the word problem translates into an

Algebraic Representation

that accurately models the situation. This representation is then manipulated through algebraic methods to

Find the Solution to the Problem

. Practice in converting word problems to algebraic equations is integral for students to excel in math and develop strong analytical and problem-solving skills.