Problem 20 - Algebra for College Students

Question

Translate each verbal sentence into an equation, using \(x\) as the variable. Then solve the equation. $$ \text { When } 75 \% \text { of a number is added to } 6 \text { , the result is } 3 \text { more than the number. Find the number. } $$

Step-by-Step Solution

Answer

The number is 12.

Step 1: Translate the verbal sentence into a mathematical equation

Start by identifying the different parts of the sentence. We know that 75% of a number is added to 6. Let the number be represented by the variable \( x \). Then 75% of the number is \( 0.75x. \) The problem also states that the result is 3 more than the number. So, the equation becomes: \[ 0.75x + 6 = x + 3. \]

Step 2: Simplify the equation

To isolate \( x \), start by getting all terms involving \( x \) on one side and constant terms on the other side. Subtract \( 0.75x \) from both sides: \[ 0.75x + 6 - 0.75x = x + 3 - 0.75x \] Simplify to get: \[ 6 = 0.25x + 3. \]

Step 3: Solve for x

Next, isolate \( x \) by first subtracting 3 from both sides: \[ 6 - 3 = 0.25x + 3 - 3 \] Simplify to get: \[ 3 = 0.25x. \] Now, divide both sides by 0.25 to solve for \( x \): \[ \frac{3}{0.25} = x \] Simplify to find: \[ x = 12. \]

Key Concepts

Translating Verbal Sentences to Equations

When dealing with verbal sentences in algebra, it helps to break down the sentence part by part. This makes it easier to convert the words into equations. Look for key phrases that indicate operations, like 'added to', 'subtracted from', 'multiplied by', and 'divided by'. For instance, in the sentence '75% of a number is added to 6, the result is 3 more than the number', we recognize several things:
1. '75% of a number' translates to \(0.75x\) if we let \(x\) be the number.
2. 'Added to 6' means we add 6 to \(0.75x\).
3. 'The result is' signals the equals sign \(=\).
4. '3 more than the number' means \(x + 3\).
Combining these parts, we get the equation: \[0.75x + 6 = x + 3.\]
The better you get at identifying these parts, the quicker you can convert verbal sentences into equations.

Solving Linear Equations

Once we have an equation, the next step is to solve it. Here, we start with: \[0.75x + 6 = x + 3.\]
Our goal is to isolate \(x\) on one side of the equation. Follow these steps:

• Move terms involving \(x\) to one side by subtracting \(0.75x\) from both sides: \[0.75x + 6 - 0.75x = x + 3 - 0.75x\]
• Simplify to get: \[6 = 0.25x + 3.\]
• Next, move the constant terms to the other side by subtracting 3 from both sides: \[6 - 3 = 0.25x + 3 - 3\]
• Simplify to get: \[3 = 0.25x.\]
• Lastly, divide both sides by \(0.25\): \[\frac{3}{0.25} = x\]
• Calculate to find: \[x = 12.\]

By isolating the variable step-by-step, you can easily find the solution to the equation.

Percentages in Algebra

Percentages often come up in algebra problems, but they can be tricky if you're not familiar with them. Here's a quick guide:

• 'Percent' means 'per hundred', so 75% is the same as 75 out of 100, which we can write as a fraction: \(\frac{75}{100}\).
• To work with percentages in algebra, convert them to decimals. For example, 75% becomes \(0.75\). This makes calculation easier.
• When a problem says '75% of a number', you multiply the number by \(0.75\). If the number is \(x\), then 75% of it is \(0.75x\).

Understanding how to handle percentages helps a lot, especially when translating word problems into equations. Get comfortable converting between percentages, fractions, and decimals to tackle these problems with confidence.