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Equations with Decimals, Fractions, and Parentheses

Distribution, collecting like terms, variables on one side

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Solve Multi-Step Equations Involving Rational Numbers
License: CC BY-NC 3.0

Carlos travelled a certain distance from home to the campgrounds on his dirt bike at a speed of 50 km/h and returned at a speed of 60 km/h. How can Carlos find the distance between his home and the campgrounds if the total time traveling was 11 hours?

In this concept, you will learn to solve multi-step equations involving rational numbers.

Rational Numbers

A rational number is any number that can be written in the form \begin{align*}\frac{a}{b}\end{align*} where ‘\begin{align*}b\end{align*}’ does not equal zero. The set of rational numbers includes integers, fractions and terminating decimals. Integers consist of all natural numbers, their opposites and zero. Any numbers from the list . . . -3, -2, -1, 0, 1, 2, 3, . . are integers. A fraction is any rational number that is not an integer. It is written as the ratio of one number to another in the form \begin{align*}\frac{a}{b}\end{align*} where ‘\begin{align*}b\end{align*}’ does not equal zero. A terminating decimal is a decimal that ends. The fraction \begin{align*}\frac{1}{8}\end{align*} expressed as a decimal is 0.125 which is a terminating decimal.

You will often have to work with these types of numbers when solving equations. Working with rational numbers when you solve an equation, does not change the rules you apply when determining the solution for the equation.

Let’s look at an example.

Solve the following equation for the variable ‘\begin{align*}x\end{align*}.’

\begin{align*}5(2x-9)+ \frac{1}{4}(-16x-8)=1\end{align*}

This equation contains both integers and fractions which are rational numbers.

First, apply the distributive property to clear the parenthesis.

\begin{align*}& 5(2x-9)+ \frac{1}{4}(-16x-8) = 1\\ & 5(2x)-5(9)+ \frac{1}{4}(-16x)+ \frac{1}{4}(-8) = 1\\ & 10x-45-4x-2 = 1\end{align*}

Next, simplify the left side of the equation.

\begin{align*}& 10x-45-4x-2 = 1\\ & 6x-47 = 1\end{align*}

Next, isolate the variable by adding 47 to both sides of the equation.

\begin{align*}& 6x-47=1\\ & 6x-47+47 =1+47\end{align*}

Next, simplify both sides of the equation.

\begin{align*}& 6x-47+47=1 + 47\\ & 6x=48\end{align*}

Then, divide both sides of the equation by 6 to solve for ‘\begin{align*}x\end{align*}.’

\begin{align*}6x &= 48\\ \frac{6x}{6} &= \frac{48}{6}\\ x &= 8\end{align*} 

The answer is 8.

Examples

Example 1

Earlier, you were given a problem about Carlos travelling on his dirt bike. He wants to figure out the distance he travelled.

First, name the variable.

Let ‘\begin{align*}d\end{align*}’ represent the distance between home and the campgrounds.

Next, use the formula \begin{align*}\text{distance}(d) = \text{velocity}(v) \times \text{time}(t)\end{align*} and rearrange the formula for finding time.

\begin{align*}d &= v \times t\\ \frac{d}{v} &= \frac{{\cancel{v}} \times t}{\cancel{v}}\\ \frac{d}{v} &= t\end{align*} 

The time of 11 hours is the time it took to go to the campground and to return home.

Next, write a verbal model to represent Carlos’ travelling.

\begin{align*}\underbrace{\text{travelling to the campgrounds}}_{\frac{d}{v}} + \underbrace{\text{travelling back to home}}_{\frac{d}{v}} = \underbrace{\text{total time travelling}}_{11}\end{align*}

Next, write and solve the equation.

\begin{align*}\frac{d}{50}+ \frac{d}{60} = 11\end{align*}

Next, simplify the left side of the equation by adding the fractions.

\begin{align*}\frac{d}{50}+ \frac{d}{60} &= 11\\ \frac{d}{50} \left(\frac{6}{6} \right)+ \frac{d}{60} \left(\frac{5}{5} \right) &= 11\\ \frac{6d}{300}+ \frac{5d}{300} &= 11\\ \frac{11d}{300} &= 11\end{align*}

Next, multiply both sides of the equation by 300.

\begin{align*}\frac{11 d}{300} &= 11\\ \overset{1}{\cancel{300}} \left(\frac{11d}{\cancel{300}} \right) &= 300(11)\\ 11d &= 3300\end{align*} 

Then, divide both sides of the equation to solve for ‘\begin{align*}d\end{align*}.’

\begin{align*}11d &= 3300\\ \frac{\overset{{1}}{\cancel{11}}d}{\cancel{11}} &= \frac{3300}{11}\\ d &= 300\end{align*}

The answer is 300.

The distance is 300 km.

Example 2

\begin{align*}5.5y-3.2(2.5y-4.6)=2.3\end{align*}

This equation contains terminating decimals which are rational numbers.

First, apply the distributive property to clear the parenthesis.

 \begin{align*}& 5.5y-3.2(2.5y-4.6) = 2.3\\ & 5.5y-3.2(2.5y)-3.2(-4.6) = 2.3\\ & 5.5y-8y+14.72 = 2.3\end{align*}

Next, simplify the left side of the equation.

\begin{align*}& 5.5y-8y+14.72=2.3\\ & -2.5y+14.72=2.3\end{align*}

Next, isolate the variable by subtracting 14.72 to both sides of the equation.

\begin{align*}& -2.5y+14.72=2.3\\ & -2.5y+14.72-14.72=2.3-14.72\end{align*}

Next, simplify both sides of the equation.

\begin{align*}& -2.5y+14.72-14.72=2.3-14.72\\ & -2.5y=-12.42\end{align*}

Then, divide both sides of the equation by -2.5 to solve for ‘\begin{align*}y\end{align*}.’

\begin{align*}-2.5y &=-12.42\\ \frac{-2.5 y}{-2.5} &=\frac{-12.42}{-2.5}\\ y &=4.968\end{align*} 

The answer is y=4.968.

Example 3

Solve the following equation for the variable:

\begin{align*}8.7n+4.5-3.2n=37.5\end{align*}

First, simplify the left side of the equation by subtracting the like terms.

\begin{align*}& 8.7n+4.5-3.2n=37.5\\ & 5.5n+4.5=37.5\end{align*}

Next, isolate the variable by subtracting 4.5 from both sides of the equation.

\begin{align*}& 5.5n+4.5=37.5\\ & 5.5n+4.5-4.5=37.5-4.5\end{align*}

Next, simplify both sides of the equation.

\begin{align*}& 5.5n+4.5-4.5=37.5-4.5\\ & 5.5n=33\end{align*}

Then, divide both sides of the equation by 5.5 to solve for ‘\begin{align*}n\end{align*}.’

\begin{align*}5.5n &= 33\\ \frac{\overset{{1}}{\cancel{5.5}}n}{\cancel{5.5}} &= \frac{33}{5.5}\\ n &= 6\end{align*}

The answer is \begin{align*}n=6\end{align*}.

Example 4

Solve the following equation for the variable:

\begin{align*}0.4k+ \frac{3}{10}+0.2k= \frac{9}{10}\end{align*}

First, simplify the left side of the equation by adding the like terms.

\begin{align*}0.4k+ \frac{3}{10}+0.2k &= \frac{9}{10}\\ 0.6k+ \frac{3}{10} &= \frac{9}{10}\end{align*}

Next, isolate the variable by subtracting \begin{align*}\frac{3}{10}\end{align*} from both sides of the equation.

\begin{align*}& 0.6k+ \frac{3}{10} = \frac{9}{10}\\ & 0.6k+ \frac{3}{10} - \frac{3}{10} = \frac{9}{10} - \frac{3}{10}\end{align*}

Next, simplify both sides of the equation.

\begin{align*}& 0.6k+ \frac{3}{10}- \frac{3}{10}=\frac{9}{10} -\frac{3}{10}\\ & 0.6k = \frac{6}{10}\end{align*}

Then, divide both sides of the equation by 0.6 to solve for ‘\begin{align*}k\end{align*}.’ However, this would involve dividing a fraction by a decimal. The division will be simpler if you either express 0.6 as a fraction \begin{align*}\left(\frac{6}{10} \right)\end{align*} or express \begin{align*}\frac{6}{10}\end{align*} as a decimal (0.6).

\begin{align*}0.6k &= \frac{6}{10}\\ 0.6k &= 0.6\\ \frac{\overset{{1}}{\cancel{0.6}}k}{\cancel{0.6}} &= \frac{\overset{{1}}{\cancel{0.6}}}{\cancel{0.6}}\\ k &= 1\end{align*}

The answer is \begin{align*}k=1\end{align*}.

Example 5

Solve the following equation for the variable:

\begin{align*}-6 \left(1- \frac{m}{12} \right) = \frac{2}{3}\end{align*} 

First, apply the distributive property to clear the parenthesis.

\begin{align*}-6 \left(1- \frac{m}{12} \right) &= \frac{2}{3}\\ -6.1- \frac{-6}{1} \cdot \frac{m}{12} &= \frac{2}{3}\\ -6+ \frac{6m}{12} &= \frac{2}{3}\end{align*}

Next, isolate the variable by adding 6 to both sides of the equation.

\begin{align*}-6+ \frac{6m}{12} &= \frac{2}{3}\\ -6+6+ \frac{6m}{12} &= \frac{2}{3}+6\end{align*}

Next, simplify both sides of the equation.

\begin{align*}-6+6+ \frac{6m}{12} &= \frac{2}{3}+6\\ \frac{6m}{12} &= \frac{2}{3} + \frac{6}{1} \left(\frac{3}{3} \right)\\ \frac{6m}{12} &= \frac{2}{3} + \frac{18}{3}\\ \frac{6m}{12} &= \frac{20}{3}\end{align*}

Next, multiply both sides of the equation by 12.

\begin{align*}\frac{6m}{12} &= \frac{20}{3}\\ \overset{1}{\cancel{12}} \left(\frac{6m}{\cancel{12}} \right) &= \overset{4}{\cancel{12}} \left(\frac{20}{\cancel{3}} \right)\\ 6m &= 80\end{align*}

Then, divide both sides of the equation by 6 to solve for ‘\begin{align*}m\end{align*}.’

\begin{align*}6m &= 80\\ \frac{\overset{{1}}{\cancel{6}}m}{\cancel{6}} &= \frac{80}{6}\\ m &= 13 \frac{2}{6}\end{align*}

Then, simplify the answer.

\begin{align*}m = 13 \frac{1}{3}\end{align*}

The answer is \begin{align*}m = 13 \frac{1}{3}\end{align*}.

Review

Solve each equation to find the value of the variable.

  1. \begin{align*}7n - 3.2n + 6.5 = 17.9\end{align*}
  2. \begin{align*}0.2(3 + p) = -5.6\end{align*}
  3. \begin{align*}s + \frac{3}{5} + \frac{1}{5} = 1\frac{2}{5}\end{align*}
  4. \begin{align*}j + \frac{5}{7} - \frac{1}{7} = 9\frac{4}{7}\end{align*}
  5. \begin{align*}\frac{3}{4} \left( g - \frac{1}{2} \right ) = \frac{1}{8}\end{align*}
  6. \begin{align*}-2 \left ( 1 - \frac{a}{4} \right ) = \frac{1}{8}\end{align*}
  7. \begin{align*}0.09y - 0.08y = .005\end{align*}
  8. \begin{align*}.28x + 4x = -8.56\end{align*}
  9. \begin{align*}\frac{1}{3}y + \frac{1}{3}y = 8\end{align*}
  10. \begin{align*}\frac{1}{4}x + \frac{1}{3} = \frac{2}{3}\end{align*}
  11. \begin{align*}\frac{1}{2}x = 18\end{align*}
  12. \begin{align*}.9x = 54\end{align*}
  13. \begin{align*}.6x + 1 = 19\end{align*}
  14. \begin{align*}\frac{1}{4}x + 2 = 19\end{align*}
  15. \begin{align*}9.05x = 27.15\end{align*}

Review (Answers)

To see the Review answers, open this PDF file and look for section 3.11. 

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Vocabulary

Decimal

In common use, a decimal refers to part of a whole number. The numbers to the left of a decimal point represent whole numbers, and each number to the right of a decimal point represents a fractional part of a power of one-tenth. For instance: The decimal value 1.24 indicates 1 whole unit, 2 tenths, and 4 hundredths (commonly described as 24 hundredths).

distributive property

The distributive property states that the product of an expression and a sum is equal to the sum of the products of the expression and each term in the sum. For example, a(b + c) = ab + ac.

fraction

A fraction is a part of a whole. A fraction is written mathematically as one value on top of another, separated by a fraction bar. It is also called a rational number.

Integer

The integers consist of all natural numbers, their opposites, and zero. Integers are numbers in the list ..., -3, -2, -1, 0, 1, 2, 3...

rational number

A rational number is a number that can be expressed as the quotient of two integers, with the denominator not equal to zero.

Repeating Decimal

A repeating decimal is a decimal number that ends with a group of digits that repeat indefinitely. 1.666... and 0.9898... are examples of repeating decimals.

Terminating Decimal

A terminating decimal is a decimal number that ends. The decimal number 0.25 is an example of a terminating decimal.

Image Attributions

  1. [1]^ License: CC BY-NC 3.0

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