# 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?

I don't see where the answers are.
Ed
Cindy and Richard leave their dorm in Charleston at the same time. Cindy rides her bicycle north at a speed of 18 miles per hour. Richard rides his bicycle south at a speed of 14 miles per hour. How long will it take them to be 96 miles apart?
3
Christopher
18t+14t=96 32t=96 32/96 3
Christopher
show that a^n-b^2n is divisible by a-b
What does 3 times your weight right now
Use algebra to combine 39×5 and the half sum of travel of 59+30
Cherokee
What is the segment of 13? Explain
Cherokee
my weight is 49. So 3 times is 147
Cherokee
kg to lbs you goin to convert 2.2 or one if the same unit your going to time your body weight by 3. example if my body weight is 210lb. what would be my weight if I was 3 times as much in kg. that's you do 210 x3 = 630lb. then 630 x 2.2= .... hope this helps
tyler
How to convert grams to pounds?
paul
What is the lcm of 340
Yes
Cherokee
How many numbers each equal to y must be taken to make 15xy
15x
Martin
15x
Asamoah
15x
Hugo
1y
Tom
1y x 15y
Tom
find the equation whose roots are 1 and 2
(x - 2)(x -1)=0 so equation is x^2-x+2=0
Ranu
I believe it's x^2-3x+2
NerdNamedGerg
because the X's multiply by the -2 and the -1 and than combine like terms
NerdNamedGerg
find the equation whose roots are -1 and 4
Ans = ×^2-3×+2
Gee
find the equation whose roots are -2 and -1
(×+1)(×-4) = x^2-3×-4
Gee
there's a chatting option in the app wow
Nana
That's cool cool
Nana
Nice to meet you all
Nana
you too.
Joan
😃
Nana
Hey you all there are several Free Apps that can really help you to better solve type Equations.
Debra
Debra, which apps specifically. ..?
Nana
am having a course in elementary algebra ,any recommendations ?
samuel
Samuel Addai, me too at ucc elementary algebra as part of my core subjects in science
Nana
me too as part of my core subjects in R M E
Ken
at ABETIFI COLLEGE OF EDUCATION
Ken
ok great. Good to know.
Joan
5x + 1/3= 2x + 1/2
sanam
Plz solve this
sanam
5x - 3x = 1/2 - 1/3 2x = 1/6 x = 1/12
Ranu
Thks ranu
sanam
a trader gains 20 rupees loses 42 rupees and then gains ten rupees Express algebraically the result of his transactions
a trader gains 20 rupees loses 42 rupees and then gains 10 rupees Express algebraically the result of his three transactions
vinaya
a trader gains 20 rupees loses 42 rupees and then gains 10 rupees Express algebraically the result of his three transactions
vinaya
a trader gains 20 rupees loses 42 rupees and then gains 10 rupees Express algebraically the result of his three transactions
vinaya
Kim is making eight gallons of punch from fruit juice and soda. The fruit juice costs $6.04 per gallon and the soda costs$4.28 per gallon. How much fruit juice and how much soda should she use so that the punch costs $5.71 per gallon? Mohamed Reply (a+b)(p+q+r)(b+c)(p+q+r)(c+a) (p+q+r) muhammad Reply 4x-7y=8 2x-7y=1 what is the answer? Ramil Reply x=7/2 & y=6/7 Pbp x=7/2 & y=6/7 use Elimination Debra true bismark factoriz e usman 4x-7y=8 X=7/4y+2 and 2x-7y=1 x=7/2y+1/2 Peggie Ok cool answer peggie Frank thanks Ramil copy and complete the table. x. 5. 8. 12. then 9x-5. to the 2nd power+4. then 2xto the second power +3x Sandra Reply What is c+4=8 Penny Reply 2 Letha 4 Lolita 4 Rich 4 thinking C+4=8 -4 -4 C =4 thinking I need to study Letha 4+4=8 William During two years in college, a student earned$9,500. The second year, she earned \$500 more than twice the amount she earned the first year.
9500=500+2x
Debra
9500-500=9000 9000÷2×=4500 X=4500
Debra
X + Y = 9500....... & Y = 500 + 2X so.... X + 500 + 2X = 9500, them X = 3000 & Y = 6500
Pbp