# 2.5 Solve equations with fractions or decimals  (Page 2/2)

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Solve: $-11=\frac{1}{2}\left(6p+2\right)$ .

$p=-4$

Solve: $8=\frac{1}{3}\left(9q+6\right)$ .

$q=2$

In the next example, even after distributing, we still have fractions to clear.

Solve: $\frac{1}{2}\left(y-5\right)=\frac{1}{4}\left(y-1\right)$ .

## Solution

 Distribute. Simplify. Multiply by the LCD, 4. Distribute. Simplify. Collect the variables to the left. Simplify. Collect the constants to the right. Simplify. Check: Let $y=9$ . Finish the check on your own.

Solve: $\frac{1}{5}\left(n+3\right)=\frac{1}{4}\left(n+2\right)$ .

$n=2$

Solve: $\frac{1}{2}\left(m-3\right)=\frac{1}{4}\left(m-7\right)$ .

$m=-1$

Solve: $\frac{5x-3}{4}=\frac{x}{2}$ .

## Solution

 Multiply by the LCD, 4. Simplify. Collect the variables to the right. Simplify. Divide. Simplify. Check: Let $x=1$ .

Solve: $\frac{4y-7}{3}=\frac{y}{6}$ .

$y=2$

Solve: $\frac{-2z-5}{4}=\frac{z}{8}$ .

$z=-2$

Solve: $\frac{a}{6}+2=\frac{a}{4}+3$ .

## Solution

 Multiply by the LCD, 12. Distribute. Simplify. Collect the variables to the right. Simplify. Collect the constants to the left. Simplify. Check: Let $a=-12$ .

Solve: $\frac{b}{10}+2=\frac{b}{4}+5$ .

$b=-20$

Solve: $\frac{c}{6}+3=\frac{c}{3}+4$ .

$c=-6$

Solve: $\frac{4q+3}{2}+6=\frac{3q+5}{4}$ .

## Solution

 Multiply by the LCD, 4. Distribute. Simplify. Collect the variables to the left. Simplify. Collect the constants to the right. Simplify. Divide by 5. Simplify. Check: Let $q=-5$ . Finish the check on your own.

Solve: $\frac{3r+5}{6}+1=\frac{4r+3}{3}$ .

$r=1$

Solve: $\frac{2s+3}{2}+1=\frac{3s+2}{4}$ .

$s=-8$

## Solve equations with decimal coefficients

Some equations have decimals in them. This kind of equation will occur when we solve problems dealing with money or percentages. But decimals can also be expressed as fractions. For example, $0.3=\frac{3}{10}$ and $0.17=\frac{17}{100}$ . So, with an equation with decimals, we can use the same method we used to clear fractions—multiply both sides of the equation by the least common denominator.

Solve: $0.06x+0.02=0.25x-1.5$ .

## Solution

Look at the decimals and think of the equivalent fractions.

$0.06=\frac{6}{100}\phantom{\rule{2em}{0ex}}0.02=\frac{2}{100}\phantom{\rule{2em}{0ex}}0.25=\frac{25}{100}\phantom{\rule{2em}{0ex}}1.5=1\frac{5}{10}$

Notice, the LCD is 100.

By multiplying by the LCD, we will clear the decimals from the equation.

 Multiply both sides by 100. Distribute. Multiply, and now we have no more decimals. Collect the variables to the right. Simplify. Collect the constants to the left. Simplify. Divide by 19. Simplify. Check: Let $x=8$ .

Solve: $0.14h+0.12=0.35h-2.4$ .

$h=12$

Solve: $0.65k-0.1=0.4k-0.35$ .

$k=-1$

The next example uses an equation that is typical of the money applications in the next chapter. Notice that we distribute the decimal before we clear all the decimals.

Solve: $0.25x+0.05\left(x+3\right)=2.85$ .

## Solution

 Distribute first. Combine like terms. To clear decimals, multiply by 100. Distribute. Subtract 15 from both sides. Simplify. Divide by 30. Simplify. Check it yourself by substituting $x=9$ into the original equation.

Solve: $0.25n+0.05\left(n+5\right)=2.95$ .

$n=9$

Solve: $0.10d+0.05\left(d-5\right)=2.15$ .

$d=16$

## Key concepts

• Strategy to Solve an Equation with Fraction Coefficients
1. Find the least common denominator of all the fractions in the equation.
2. Multiply both sides of the equation by that LCD. This clears the fractions.
3. Solve using the General Strategy for Solving Linear Equations.

## Practice makes perfect

Solve Equations with Fraction Coefficients

In the following exercises, solve each equation with fraction coefficients.

$\frac{1}{4}x-\frac{1}{2}=-\frac{3}{4}$

$\frac{3}{4}x-\frac{1}{2}=\frac{1}{4}$

$x=1$

$\frac{5}{6}y-\frac{2}{3}=-\frac{3}{2}$

$\frac{5}{6}y-\frac{1}{3}=-\frac{7}{6}$

$y=-1$

$\frac{1}{2}a+\frac{3}{8}=\frac{3}{4}$

$\frac{5}{8}b+\frac{1}{2}=-\frac{3}{4}$

$b=-2$

$2=\frac{1}{3}x-\frac{1}{2}x+\frac{2}{3}x$

$2=\frac{3}{5}x-\frac{1}{3}x+\frac{2}{5}x$

$x=3$

$\frac{1}{4}m-\frac{4}{5}m+\frac{1}{2}m=-1$

$\frac{5}{6}n-\frac{1}{4}n-\frac{1}{2}n=-2$

$n=-24$

$x+\frac{1}{2}=\frac{2}{3}x-\frac{1}{2}$

$x+\frac{3}{4}=\frac{1}{2}x-\frac{5}{4}$

$x=-4$

$\frac{1}{3}w+\frac{5}{4}=w-\frac{1}{4}$

$\frac{3}{2}z+\frac{1}{3}=z-\frac{2}{3}$

$z=-2$

$\frac{1}{2}x-\frac{1}{4}=\frac{1}{12}x+\frac{1}{6}$

$\frac{1}{2}a-\frac{1}{4}=\frac{1}{6}a+\frac{1}{12}$

$a=1$

$\frac{1}{3}b+\frac{1}{5}=\frac{2}{5}b-\frac{3}{5}$

$\frac{1}{3}x+\frac{2}{5}=\frac{1}{5}x-\frac{2}{5}$

$x=-6$

$1=\frac{1}{6}\left(12x-6\right)$

$1=\frac{1}{5}\left(15x-10\right)$

$x=1$

$\frac{1}{4}\left(p-7\right)=\frac{1}{3}\left(p+5\right)$

$\frac{1}{5}\left(q+3\right)=\frac{1}{2}\left(q-3\right)$

$q=7$

$\frac{1}{2}\left(x+4\right)=\frac{3}{4}$

$\frac{1}{3}\left(x+5\right)=\frac{5}{6}$

$x=-\frac{5}{2}$

$\frac{5q-8}{5}=\frac{2q}{10}$

$\frac{4m+2}{6}=\frac{m}{3}$

$m=-1$

$\frac{4n+8}{4}=\frac{n}{3}$

$\frac{3p+6}{3}=\frac{p}{2}$

$p=-4$

$\frac{u}{3}-4=\frac{u}{2}-3$

$\frac{v}{10}+1=\frac{v}{4}-2$

$v=20$

$\frac{c}{15}+1=\frac{c}{10}-1$

$\frac{d}{6}+3=\frac{d}{8}+2$

$d=-24$

$\frac{3x+4}{2}+1=\frac{5x+10}{8}$

$\frac{10y-2}{3}+3=\frac{10y+1}{9}$

$y=-1$

$\frac{7u-1}{4}-1=\frac{4u+8}{5}$

$\frac{3v-6}{2}+5=\frac{11v-4}{5}$

$v=4$

Solve Equations with Decimal Coefficients

In the following exercises, solve each equation with decimal coefficients.

$0.6y+3=9$

$0.4y-4=2$

$y=15$

$3.6j-2=5.2$

$2.1k+3=7.2$

$k=2$

$0.4x+0.6=0.5x-1.2$

$0.7x+0.4=0.6x+2.4$

$x=20$

$0.23x+1.47=0.37x-1.05$

$0.48x+1.56=0.58x-0.64$

$x=22$

$0.9x-1.25=0.75x+1.75$

$1.2x-0.91=0.8x+2.29$

$x=8$

$0.05n+0.10\left(n+8\right)=2.15$

$0.05n+0.10\left(n+7\right)=3.55$

$n=19$

$0.10d+0.25\left(d+5\right)=4.05$

$0.10d+0.25\left(d+7\right)=5.25$

$d=10$

$0.05\left(q-5\right)+0.25q=3.05$

$0.05\left(q-8\right)+0.25q=4.10$

$q=15$

## Everyday math

Coins Taylor has $2.00 in dimes and pennies. The number of pennies is 2 more than the number of dimes. Solve the equation $0.10d+0.01\left(d+2\right)=2$ for $d$ , the number of dimes. Stamps Paula bought$22.82 worth of 49-cent stamps and 21-cent stamps. The number of 21-cent stamps was 8 less than the number of 49-cent stamps. Solve the equation $0.49s+0.21\left(s-8\right)=22.82$ for s , to find the number of 49-cent stamps Paula bought.

$s=35$

## Writing exercises

Explain how you find the least common denominator of $\frac{3}{8}$ , $\frac{1}{6}$ , and $\frac{2}{3}$ .

If an equation has several fractions, how does multiplying both sides by the LCD make it easier to solve?

If an equation has fractions only on one side, why do you have to multiply both sides of the equation by the LCD?

In the equation $0.35x+2.1=3.85$ what is the LCD? How do you know?

100. Justifications will vary.

## Self check

After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

Overall, after looking at the checklist, do you think you are well-prepared for the next section? Why or why not?

Washing his dad’s car alone, eight-year-old Levi takes 2.5 hours. If his dad helps him, then it takes 1 hour. How long does it take the Levi’s dad to wash the car by himself?
Ethan and Leo start riding their bikes at the opposite ends of a 65-mile bike path. After Ethan has ridden 1.5 hours and Leo has ridden 2 hours, they meet on the path. Ethan’s speed is 6 miles per hour faster than Leo’s speed. Find the speed of the two bikers.
Nathan walked on an asphalt pathway for 12 miles. He walked the 12 miles back to his car on a gravel road through the forest. On the asphalt he walked 2 miles per hour faster than on the gravel. The walk on the gravel took one hour longer than the walk on the asphalt. How fast did he walk on the gravel?
Mckenzie
Nancy took a 3 hour drive. She went 50 miles before she got caught in a storm. Then she drove 68 miles at 9 mph less than she had driven when the weather was good. What was her speed driving in the storm?
Mr Hernaez runs his car at a regular speed of 50 kph and Mr Ranola at 36 kph. They started at the same place at 5:30 am and took opposite directions. At what time were they 129 km apart?
90 minutes
Melody wants to sell bags of mixed candy at her lemonade stand. She will mix chocolate pieces that cost $4.89 per bag with peanut butter pieces that cost$3.79 per bag to get a total of twenty-five bags of mixed candy. Melody wants the bags of mixed candy to cost her $4.23 a bag to make. How many bags of chocolate pieces and how many bags of peanut butter pieces should she use? Jake Reply enrique borrowed$23,500 to buy a car he pays his uncle 2% interest on the $4,500 he borrowed from him and he pays the bank 11.5% interest on the rest. what average interest rate does he pay on the total$23,500
13.5
Pervaiz
Amber wants to put tiles on the backsplash of her kitchen counters. She will need 36 square feet of tiles. She will use basic tiles that cost $8 per square foot and decorator tiles that cost$20 per square foot. How many square feet of each tile should she use so that the overall cost of the backsplash will be $10 per square foot? Bridget Reply The equation P=28+2.54w models the relation between the amount of Randy’s monthly water bill payment, P, in dollars, and the number of units of water, w, used. Find the payment for a month when Randy used 15 units of water. Bridget help me understand graphs Marlene Reply what kind of graphs? bruce function f(x) to find each value Marlene I am in algebra 1. Can anyone give me any ideas to help me learn this stuff. Teacher and tutor not helping much. Marlene Given f(x)=2x+2, find f(2) so you replace the x with the 2, f(2)=2(2)+2, which is f(2)=6 Melissa if they say find f(5) then the answer would be f(5)=12 Melissa I need you to help me Melissa. Wish I can show you my homework Marlene How is f(1) =0 I am really confused Marlene what's the formula given? f(x)=? Melissa It shows a graph that I wish I could send photo of to you on here Marlene Which problem specifically? Melissa which problem? Melissa I don't know any to be honest. But whatever you can help me with for I can practice will help Marlene I got it. sorry, was out and about. I'll look at it now. Melissa Thank you. I appreciate it because my teacher assumes I know this. My teacher before him never went over this and several other things. Marlene I just responded. Melissa Thank you Marlene -65r to the 4th power-50r cubed-15r squared+8r+23 ÷ 5r WENDY Reply State the question clearly please Rich write in this form a/b answer should be in the simplest form 5% August Reply convert to decimal 9/11 August 0.81818 Rich 5/100 = .05 but Rich is right that 9/11 = .81818 Melissa Equation in the form of a pending point y+2=1/6(×-4) Jose Reply write in simplest form 3 4/2 August definition of quadratic formula Ahmed Reply From Google: The quadratic formula, , is used in algebra to solve quadratic equations (polynomial equations of the second degree). The general form of a quadratic equation is , where x represents a variable, and a, b, and c are constants, with . A quadratic equation has two solutions, called roots. Melissa what is the answer of w-2.6=7.55 What Reply 10.15 Michael w = 10.15 You add 2.6 to both sides and then solve for w (-2.6 zeros out on the left and leaves you with w= 7.55 + 2.6) Korin Nataly is considering two job offers. The first job would pay her$83,000 per year. The second would pay her $66,500 plus 15% of her total sales. What would her total sales need to be for her salary on the second offer be higher than the first? Mckenzie Reply x >$110,000
bruce
greater than $110,000 Michael Estelle is making 30 pounds of fruit salad from strawberries and blueberries. Strawberries cost$1.80 per pound, and blueberries cost $4.50 per pound. If Estelle wants the fruit salad to cost her$2.52 per pound, how many pounds of each berry should she use?
$1.38 worth of strawberries +$1.14 worth of blueberries which= $2.52 Leitha how Zaione is it right😊 Leitha lol maybe Robinson 8 pound of blueberries and 22 pounds of strawberries Melissa 8 pounds x 4.5 = 36 22 pounds x 1.80 = 39.60 36 + 39.60 = 75.60 75.60 / 30 = average 2.52 per pound Melissa 8 pounds x 4.5 equal 36 22 pounds x 1.80 equal 39.60 36 + 39.60 equal 75.60 75.60 / 30 equal average 2.52 per pound Melissa hmmmm...... ? Robinson 8 pounds x 4.5 = 36 22 pounds x 1.80 = 39.60 36 + 39.60 = 75.60 75.60 / 30 = average 2.52 per pound Melissa The question asks how many pounds of each in order for her to have an average cost of$2.52. She needs 30 lb in all so 30 pounds times $2.52 equals$75.60. that's how much money she is spending on the fruit. That means she would need 8 pounds of blueberries and 22 lbs of strawberries to equal 75.60
Melissa
good
Robinson
👍
Leitha
thanks Melissa.
Leitha
nawal let's do another😊
Leitha
we can't use emojis...I see now
Leitha
Sorry for the multi post. My phone glitches.
Melissa