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Chapter 6: Problem 267

In the following exercises, solve each proportion. $$ \frac{56}{72}=\frac{y}{9} $$

Short Answer

Expert verified
The value of \( y \) is 7.

Step by step solution

01

- Set Up the Proportion

First, rewrite the given proportion: \[ \frac{56}{72} = \frac{y}{9} \]
02

- Cross Multiply

To solve for the variable \( y \), cross multiply the terms: \[ 56 \times 9 = 72 \times y \]
03

- Calculate the Left Side

Compute \( 56 \times 9 \): \[ 56 \times 9 = 504 \]
04

- Solve for \( y \)

Now set up the equation from the cross multiplication step and solve for \( y \): \[ 504 = 72y \] Divide both sides by 72 to isolate \( y \): \[ y = \frac{504}{72} \] \[ y = 7 \]

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Key Concepts

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

Cross Multiplication
In solving proportions, cross multiplication is a fundamental technique. When you have a proportion equation, it's easy to feel overwhelmed by the fractions. But cross-multiplying simplifies it all.
Cross-multiplication involves multiplying the numerator of one fraction by the denominator of the other fraction. By doing this for both sides of the equation, you eliminate the fractions and end up with a simple multiplication statement. For example:
Given the proportion \(\frac{56}{72} = \frac{y}{9}\):
1. Cross multiply: 56 \(\times\) 9 = 72 \(\times\) y.
2. Simplify: 504 = 72y.
This process reduces confusion and makes it easier to solve for the variable.
Proportion Equations
Proportion equations compare two ratios or fractions and set them equal to each other. These equations are handy because they allow you to solve for unknown values in real-world scenarios.
In the given example, the goal is to determine the unknown value \(y\) in the proportion \(\frac{56}{72} = \frac{y}{9}\).
To solve, follow these steps:
  • Set up the proportion clearly and ensure all parts are correctly placed.

  • Cross-multiply to eliminate the fractions.

  • Solve the resulting equation to find the unknown value.

Proportions are useful in various contexts, such as converting units, scaling recipes, and interpreting maps.
Basic Algebra
Understanding basic algebra is essential for solving equations, like the one in this exercise. Algebra involves using symbols, typically letters, to represent unknown values and creating equations to solve for these values.
The process usually includes:
  • Identifying the unknown variable (e.g., \(y\) in our exercise).

  • Performing operations to isolate the variable, such as adding, subtracting, multiplying, or dividing both sides of the equation.

Let's break down the steps again using our previous example:
1. Set the equation after cross-multiplying: 504 = 72y.
2. Isolate \(y\) by dividing both sides by 72: \(\frac{504}{72}\).
3. Calculate to find \(y\): \(y = 7\).
By mastering these basics, algebraic problems become much less intimidating.

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Most popular questions from this chapter

In the following exercises, write each sentence as a proportion. $$ \frac{11.6}{9.2}=\frac{37.12}{29.44} $$

In the following exercises, solve the percent problem. Kiyoshi bought a television set on sale for \(\$ 899\) . The original price was \(\$ 1,200\) . Find: (a) the amount of discount (b) the discount rate (round to the nearest tenth of a percent)

In the following exercises, convert each percent to (a) a decimal (b) a simplified fraction. $$ 24 \% $$

In the following exercises, solve the proportion problem. The children's dosage of acetaminophen is 5 milliliters (ml) for every 25 pounds of a child's weight. How many milliters of acetaminophen will be prescribed for a 60 pound child?

In the following exercises, solve the simple interest problem. Find the principal invested if \(\$ 660\) interest was earned in 5 years at an interest rate of 3\(\%\) .
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