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Chapter 5: Problem 18

Solve. $$ \frac{2}{24}=\frac{x}{36} $$

Short Answer

Expert verified
x = 3

Step by step solution

01

Simplify the Fraction

Simplify the fraction on the left side of the equation: \( \frac{2}{24} = \frac{1}{12} \). To do this, divide both the numerator and the denominator by their greatest common divisor, which is 2.
02

Set Up the Proportion

Rewrite the equation with the simplified fraction: \( \frac{1}{12}=\frac{x}{36} \). This keeps the proportion intact.
03

Cross-Multiply

Cross-multiply to solve for \( x \). Multiply 1 by 36 and 12 by \( x \): \( 1 \times 36 = 12x \).
04

Solve for \( x \)

To find \( x \), divide both sides of the equation by 12: \( x = \frac{36}{12} = 3 \).

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Key Concepts

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

Solving Proportions
A proportion is an equation that states two ratios are equal. In this exercise, we are given the proportion: `$$ \frac{2}{24}=\frac{x}{36} $$`. The goal is to find the value of `x` that makes the two ratios equal. To solve proportions, you can follow some straightforward steps:
  • Simplify the fractions to make the calculations easier.
  • Use cross-multiplication.
  • Solve for the remaining variable.
This method is useful because it allows us to solve for unknown values quickly and accurately.
Cross-Multiplication
Cross-multiplication is a technique used to solve proportions. It involves multiplying the numerator of one fraction by the denominator of the other. Let's see how it's done in this exercise. After simplifying the fraction `$$ \frac{2}{24} $$` to `$$ \frac{1}{12} $$`, we rewrite the proportion: `
$$ \frac{1}{12}=\frac{x}{36}$$ `. Next, we apply the cross-multiplication rule:
`1 脳 36 = 12 脳 x`. Simplifying this equation, we get `36 = 12x`. This equation is much easier to work with and leads us directly to the solution for `x` when we divide both sides by 12.
Simplifying Fractions
Simplifying fractions involves reducing them to their simplest form. This means finding the greatest common divisor (GCD) of the numerator and the denominator and dividing both by this number. For instance, in the exercise, the given fraction is `$$ \frac{2}{24} $$. The GCD of 2 and 24 is 2, so we divide both the numerator and the denominator by 2:
$$ \frac{2}{24} $$ becomes `$$ \frac{1}{12} $$`. Simplifying fractions before solving proportions helps to make the numbers smaller and more manageable. This step makes the overall problem easier and reduces the effort involved in calculations.

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