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Chapter 4: Problem 18

In Exercises \(11-18,\) solve each proportion. $$\frac{y}{18}=\frac{10}{9}$$

Short Answer

Expert verified
\( y = 20 \)

Step by step solution

01

- Understand the Given Proportion

The given proportion is \(\frac{y}{18} = \frac{10}{9}\). This means that two ratios are equal.
02

- Cross Multiply

To solve for \(y\), cross-multiply the terms of the proportion. This means multiplying the numerator of the first fraction by the denominator of the second fraction and setting it equal to the numerator of the second fraction multiplied by the denominator of the first fraction: \(y \times 9 = 18 \times 10\).
03

- Simplify the Equation

Perform the multiplication on both sides of the equation. This gives \(9y = 180\).
04

- Solve for y

To isolate \(y\), divide both sides of the equation by 9: \(y = \frac{180}{9}\). Simplify the fraction to get \(y = 20\).

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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 or equations that involve fractions. In a proportion such as \(\frac{y}{18} = \frac{10}{9}\), we see that two ratios are set equal. Cross multiplication helps us find the unknown variable by eliminating the fractions. To cross-multiply, we multiply the denominator of one fraction by the numerator of the other fraction and set the products equal to each other. For example:
\(\frac{y}{18} = \frac{10}{9}\) becomes \(y \times 9 = 18 \times 10\)
This will transform the proportion into a simpler algebraic equation without fractions, which is easier to solve.
Ratios
A ratio compares two quantities showing the relative sizes of each. Mathematically it is expressed as \(a:b\) or as a fraction \(\frac{a}{b}\). In proportions, two ratios are set equal to express that they represent the same relationship:
\(\frac{y}{18} = \frac{10}{9}\)
This tells us that \(y\) and 18 have the same relationship as 10 and 9. Simplifying ratios can often help to understand or solve for unknowns. In solving the given proportion, we recognize the equality of these relationships which allows us to use cross multiplication to find the value of \(y\).
Algebraic Equations
Algebraic equations are mathematical statements showing the equality of two expressions. When solving proportions through cross multiplication, we convert the proportion into an algebraic equation:
From \(y \times 9 = 18 \times 10\), we simplify to get \(9y = 180\). This is an algebraic equation where \(y\) is the unknown variable. To solve for \(y\), we isolate it on one side by dividing both sides of the equation by 9:
  • The equation becomes \(\frac{9y}{9} = \frac{180}{9}\)
  • This simplifies to \(y = 20\)
With \(y\) isolated, we find the solution to the proportion. Algebraic equations are powerful tools because they can be manipulated using various operations to find unknown values.

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Most popular questions from this chapter

Set up an equation or inequality and solve the problem. Be sure to indicate clearly what quantity your variable represents. Round to the nearest tenth where necessary. A total of \(\$ 5000\) is to be invested, part at \(8 \%\) and the remainder at \(10 \% .\) If the interest from the two investments is to be at least \(\$ 460\), what is the least that must be invested at \(10 \% ?\)

Solve the equations and inequalities. $$0.155 x+0.075(80-x)=8.4$$

Use a calculator to do the computation. Round off to three decimal places where necessary. $$\frac{12}{7} \cdot \frac{5}{8}$$

In each of the following exercises, perform the indicated operations. Express your answer as a single fraction reduced to lowest terms. $$\frac{2 x+3}{x}+\frac{x}{2}$$

Use a calculator to do the computation. Round off to three decimal places where necessary. $$\frac{5}{6} \div \frac{3}{4}$$
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