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Chapter 6: Problem 11

Write each rate in lowest terms. Tyler earned \(\$ 58\) in 8 hr.

Short Answer

Expert verified
\[\frac{29}{4}\] dollars per hour.

Step by step solution

01

Write the rate as a fraction

Express the given rate as a fraction where the numerator represents the amount earned and the denominator represents the time taken. Thus, the fraction is \[\frac{58}{8}\frac{\text{dollars}}{\text{hours}} \].
02

Simplify the fraction

To simplify the fraction, find the greatest common divisor (GCD) of 58 and 8. The GCD of 58 and 8 is 2. Divide both the numerator and the denominator by their GCD: \[\frac{58 \/ 2}{8 \/ 2} = \frac{29}{4} \].
03

Write the simplified rate

The simplified rate is now \[\frac{29}{4}\] dollars per hour.

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Key Concepts

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

Fraction Reduction
When we simplify a rate expressed as a fraction, we are performing a process known as fraction reduction. This involves reducing the fraction to its simplest form so that it's easier to interpret and work with.

Here's how to do it:
  • Write the rate as a fraction, with the numerator and the denominator.
  • Find a common factor for both the numerator and the denominator.
  • Divide both by this factor to get the simplest form of the fraction.

In our example, we started with the fraction \(\frac{58}{8}\) (dollars per hour). By identifying a common divisor (2) for both 58 and 8, we divide both numbers, reducing the fraction to \(\frac{29}{4}\). This is a simpler and cleaner way to represent the rate.

Simplifying fractions makes comparing rates and performing calculations much more manageable.
Greatest Common Divisor
One of the key steps in fraction reduction is finding the greatest common divisor (GCD), also known as the greatest common factor (GCF). This is the largest number that divides both the numerator and the denominator without leaving a remainder.

To find the GCD of two numbers, you can use different methods:
  • Prime Factorization: Break down both numbers into their prime factors and multiply the common factors.
  • Euclidean Algorithm: This is an efficient method that involves division and taking remainders.

In our example, the numbers 58 and 8 have a GCD of 2. By dividing both 58 (numerator) and 8 (denominator) by 2, we simplify the fraction to \(\frac{29}{4}\).

Understanding how to find the GCD helps in reducing fractions more efficiently and accurately.
Dollars per Hour
The term 'dollars per hour' refers to a rate that measures how much money is earned for each hour of work. This type of rate is commonly used to describe wages or earnings.

Consider the rate \(\frac{58}{8}\) dollars per hour, which means \(58 earned over 8 hours.
  • To simplify this rate, we simplify the fraction to \(\frac{29}{4}\), making it 29 dollars for every 4 hours of work.
  • This can be understood as \)7.25 per hour (since \(\frac{29}{4} = 7.25\)).

Simplifying rates like this helps you quickly understand and compare different earnings or wage scenarios. It provides clarity on how much is made per hour, which is essential for budgeting, job comparisons, and financial planning.

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Most popular questions from this chapter

A person \(1.6 \mathrm{m}\) tall stands next to a lifeguard observation platform. If the person casts a shadow of \(1 \mathrm{m}\) and the lifeguard platform casts a shadow of \(1.5 \mathrm{m},\) how high is the platform?

Determine whether the given value is a solution to the proportion. $$\frac{14}{x}=\frac{12}{-18}, \quad x=-21$$

Refer to the table that shows the average spending per person for reading (books, newspapers, magazines, etc.) by age group. Write each ratio in lowest terms. (PICTURE NOT COPY) \begin{array}{|l|c|} \hline \text { Age Group } & \text { Annual Average (\$) } \\ \hline \text { Under 25 years } & 60 \\ \hline 25 \text { to } 34 \text { years } & 111 \\ \hline 35 \text { to } 44 \text { years } & 136 \\ \hline 45 \text { to } 54 \text { years } & 172 \\ \hline 55 \text { to } 64 \text { years } & 183 \\ \hline 65 \text { to } 74 \text { years } & 159 \\ \hline 75 \text { years and over } & 128 \\ \hline \end{array} Find the ratio of spending for the group 35 to 44 years old to the spending for the group 45 to 54 years old.

Determine whether the ratios form a proportion. $$\frac{5}{18}=\frac{4}{16}$$

Solve the proportion. $$\frac{p}{9}=\frac{1}{6}$$
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