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Chapter 5: Problem 46

Solve. $$ \frac{10.4}{12.4}=\frac{6.76}{t} $$

Short Answer

Expert verified
t = 8.06

Step by step solution

01

Understand the Proportion Problem

The equation given is a proportion: \( \frac{10.4}{12.4} = \frac{6.76}{t} \). This means that the ratios on both sides are equal. We need to find the value of \( t \) that makes this proportion true.
02

Cross-Multiply

To solve for \( t \), use cross-multiplication. This involves multiplying the numerator of one fraction by the denominator of the other fraction: \( 10.4 \times t = 12.4 \times 6.76 \)
03

Simplify the Equation

Calculate the right-hand side of the equation: \( 12.4 \times 6.76 = 83.824 \) So the equation becomes: \( 10.4t = 83.824 \)
04

Solve for \( t \)

To isolate \( t \), divide both sides of the equation by 10.4: \( t = \frac{83.824}{10.4} \)Perform the division to find the value of \( t \): \( t = 8.06 \)

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Key Concepts

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

cross-multiplication
Cross-multiplication is a method used to solve proportions, which are equations stating that two ratios are equal. In this exercise, the given proportion is \( \frac{10.4}{12.4} = \frac{6.76}{t} \). Cross-multiplication involves multiplying the numerator of one fraction by the denominator of the other fraction and equating the products. For our given equation, you multiply 10.4 by \( t \) and 12.4 by 6.76. This gives us:

\( 10.4 \times t = 12.4 \times 6.76 \).

This step converts the proportion into a standard algebraic equation, which simplifies the process of solving for the unknown variable.
solving equations
Once you have an equation from cross-multiplication, the next step is to solve for the unknown variable. From our earlier example, we have:
\( 10.4t = 12.4 \times 6.76 \).

First, calculate \(12.4 \times 6.76\). Performing the multiplication gives:
\( 12.4 \times 6.76 = 83.824 \).

So, the equation updates to:
\( 10.4t = 83.824 \).

At this point, we need to isolate \( t \). To do so, we must perform a division, which brings us to the next core concept.
division
To isolate \( t \) in the equation \( 10.4t = 83.824 \), you divide both sides by the coefficient of \( t \), which is 10.4. This step looks like:
\( t = \frac{83.824}{10.4} \).

By performing this division, we get:
\( t = 8.06 \).

Therefore, the value of \( t \) that satisfies the original proportion \( \frac{10.4}{12.4} = \frac{6.76}{t} \) is 8.06.
Division is essentially spreading out a given quantity into equal parts, and in our context, it ensures we find the precise value of our unknown variable by balancing the equation.

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