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Chapter 5: Problem 41

Solve. $$ \frac{t}{0.16}=\frac{0.15}{0.40} $$

Short Answer

Expert verified
t = 0.06

Step by step solution

01

Understand the equation

You are given the equation \( \frac{t}{0.16} = \frac{0.15}{0.40} \). The goal is to solve for the variable \( t \).
02

Simplify the right side

Simplify the fraction on the right side of the equation: \( \frac{0.15}{0.40} \). Divide the numerator and denominator by their greatest common divisor, which is 0.05. Thus, \( \frac{0.15}{0.40} = \frac{0.15 \/ 0.05}{0.40 \/ 0.05} = \frac{3}{8} \). So the equation becomes \( \frac{t}{0.16} = \frac{3}{8} \).
03

Cross multiply to solve for t

Cross multiply to solve for \( t \): \( t \cdot 8 = 0.16 \cdot 3 \). This simplifies to \( 8t = 0.48 \).
04

Isolate t

To isolate \( t \), divide both sides of the equation by 8: \( t = \frac{0.48}{8} \).
05

Simplify the final value

Simplify \( \frac{0.48}{8} \) to find the value of \( t \): \( t = 0.06 \). Thus, \( t = 0.06 \).

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Key Concepts

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

Cross Multiplication
To solve the equation \( \frac{t}{0.16} = \frac{0.15}{0.40} \), we employ a technique called cross multiplication. This helps in isolating the variable. In cross multiplication, you multiply the numerator of one fraction by the denominator of the other. For the given equation, we multiply \( t \) by 8 and 0.16 by 3. This creates a new and simpler equation: \( 8t = 0.48 \). Cross multiplication thus transforms the equation into a more manageable form, enabling us to solve for the variable.
Simplification of Fractions
Before we can use cross multiplication, we simplify the fraction \( \frac{0.15}{0.40} \). Simplifying fractions makes the numbers smaller and easier to manage. To do this, we divide both the numerator (0.15) and the denominator (0.40) by their greatest common divisor (GCD). In this case, the GCD is 0.05. Dividing both by 0.05, the fraction simplifies to \( \frac{3}{8} \). Simplified fractions are crucial as they pave the way for easier calculations in equations.
Division
After cross multiplying, we need to isolate the variable \( t \). The resulting equation from cross multiplication is \( 8t = 0.48 \). To find \( t \), we perform a division operation. We divide 0.48 by 8. The equation \( t = \frac{0.48}{8} \) leads to \( t = 0.06 \) after performing the division. Division helps in isolating the variable we are solving for, making it the final step in the solution process.

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