Verified Answer :

  • For the ratio \(2 : 3 = 6 : x\), we solve \(2x = 18\) to find \(x = 9\).
  • For the ratio \(4 : 7 = x : 42\), we solve \(7x = 168\) to find \(x = 24\).
  • For the ratio \(2x : 48 = 3 : 12\), we solve \(2x = 12\) to find \(x = 6\).
  • For the ratio \(12 : 20 = x : 15\), we solve \(x = 9\).
  • Final values: \(\boxed{x = 9, 24, 6, 9}\).

Explanation

  1. Problem Analysis

We are given four proportional equations where a value \(x\) is unknown. To find \(x\), we use the property of proportions: for any ratio \(a : b = c : d\), it is equivalent to the equation \(\frac{a}{b} = \frac{c}{d}\), which can be solved using cross-multiplication: \(a \times d = b \times c\). We will solve each equation individually.

  1. Solving Equation 1

For the first ratio \(2 : 3 = 6 : x\), we set up the fraction \(\frac{2}{3} = \frac{6}{x}\). Using cross-multiplication, we have \(2 \times x = 3 \times 6\), which simplifies to \(2x = 18\). Dividing both sides by 2, we get \(x = \frac{18}{2} = 9\).

  1. Solving Equation 2

For the second ratio \(4 : 7 = x : 42\), we set up the fraction \(\frac{4}{7} = \frac{x}{42}\). Cross-multiplying gives \(7 \times x = 4 \times 42\). We compute \(4 \times 42 = 168\), so \(7x = 168\). Dividing by 7, we find \(x = \frac{168}{7} = 24\).

  1. Solving Equation 3

For the third ratio \(2x : 48 = 3 : 12\), we set up the fraction \(\frac{2x}{48} = \frac{3}{12}\). Simplifying the right side, \(\frac{3}{12} = \frac{1}{4}\), so \(\frac{2x}{48} = \frac{1}{4}\). Multiplying both sides by 48, we get \(2x = \frac{48}{4} = 12\). Dividing by 2 gives \(x = \frac{12}{2} = 6\).

  1. Solving Equation 4

For the fourth ratio \(12 : 20 = x : 15\), we set up the fraction \(\frac{12}{20} = \frac{x}{15}\). Simplifying the left side, \(\frac{12}{20} = \frac{3}{5}\), so \(\frac{3}{5} = \frac{x}{15}\). Multiplying both sides by 15, we get \(x = 15 \times \frac{3}{5} = 3 \times 3 = 9\).

Examples

Proportions are essential in everyday life, such as when you are cooking and need to scale a recipe. For instance, if a recipe calls for 2 cups of flour for every 3 cups of milk (a ratio of \(2:3\)), and you only have 6 cups of flour, you can use the proportion \(2/3 = 6/x\) to determine that you need \(x = 9\) cups of milk. Understanding these relationships helps ensure ingredients, construction materials, and even time management are handled accurately and proportionally.

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