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Chapter 7: Problem 257

Solve each proportion. $$\frac{a}{a+12}=\frac{4}{7}$$

Short Answer

Expert verified
a = 16

Step by step solution

01

- Set up the Proportion

Given the proportion \(\frac{a}{a+12} = \frac{4}{7}\), set up the equation by cross-multiplying to eliminate the fractions.
02

- Cross-Multiply

Cross-multiplying the terms gives us: \(a \times 7 = 4 \times (a+12)\).This results in: \(7a = 4(a + 12)\).
03

- Distribute

Distribute the 4 on the right-hand side: \(7a = 4a + 48\).
04

- Solve for a

Subtract \(4a\) from both sides to isolate the variable: \(7a - 4a = 48\), which simplifies to: \(3a = 48\).
05

- Divide by the Coefficient

Divide both sides by 3 to solve for \(a\): \(a = \frac{48}{3}\), which simplifies to: \(a = 16\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Cross-Multiplication
When we solve proportions, cross-multiplication is a handy method. It allows us to eliminate fractions, making our equations easier to handle.
Let's look at the given proportion: \frac{a}{a + 12} = \frac{4}{7}\.
To cross-multiply, we multiply the numerator of one fraction by the denominator of the other fraction:
  • Left side: \(a \times 7 = 7a\)
  • Right side: \(4 \times (a + 12)\)
This results in: \(7a = 4(a + 12)\).
With cross-multiplying, we now have an equation without fractions. This greatly simplifies the process of solving proportions.
Distributive Property
The distributive property helps us get rid of parentheses in our equations. Applying it means we distribute multiplication over addition or subtraction inside parentheses.
From our previous step, we have the equation: \(7a = 4(a + 12)\).
To remove the parentheses, we distribute the 4:
  • \(4 \times a = 4a\)
  • \(4 \times 12 = 48\)
This transforms our equation into: \(7a = 4a + 48\).
Using the distributive property here helps break down the equation and makes it simpler to isolate the variable in the next steps.
Isolating Variables
Isolating the variable means getting the variable alone on one side of the equation. This allows us to solve for that variable's value.
From our last step, we have: \(7a = 4a + 48\).
To isolate \(a\), we need to get rid of the \(4a\) on the right side. We do this by subtracting \(4a\) from both sides:
  • \(7a - 4a = 3a\)
  • \(4a + 48 - 4a = 48\)
Now we have: \(3a = 48\).
Finally, divide both sides by 3 to solve for \(a\):
  • \(\frac{3a}{3} = a\)
  • \(\frac{48}{3} = 16\)
Thus, \(a = 16\).
Following these steps helps us isolate the variable and find its value efficiently.

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Most popular questions from this chapter

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