# 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?

In 10 years, the population of Detroit fell from 950,000 to about 712,500. Find the percent decrease.
how do i set this up
Jenise
25%
Melissa
25 percent
Muzamil
950,000 - 712,500 = 237,500. 237,500 / 950,000 = .25 = 25%
Melissa
I've tried several times it won't let me post the breakdown of how you get 25%.
Melissa
Subtract one from the other to get the difference. Then take that difference and divided by 950000 and you will get .25 aka 25%
Melissa
Finally 👍
Melissa
one way is to set as ratio: 100%/950000 = x% / 712500, which yields that 712500 is 75% of the initial 950000. therefore, the decrease is 25%.
bruce
twenty five percent...
Jeorge
thanks melissa
Jeorge
950000-713500 *100 and then divide by 950000 = 25
Muzamil
Jeannette has $5 and$10 bills in her wallet. The number of fives is three more than six times the number of tens. Let t represent the number of tens. Write an expression for the number of fives.
6t+3
Melissa
6t +3
Bollywood
Tricia got a 6% raise on her weekly salary. The raise was $30 per week. What was her original salary? Iris Reply let us suppose her original salary is 'm'. so, according to the given condition, m*(6/100)=30 m= (30*100)/6 m= 500 hence, her original salary is$500.
Simply
28.50
Toi
thanks
Jeorge
How many pounds of nuts selling for $6 per pound and raisins selling for$3 per pound should Kurt combine to obtain 120 pounds of trail mix that cost him $5 per pound? Valeria Reply 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 code $20 per square foot. How many square feet of each tile should she use so that the overal cost of he backsplash will be$10 per square foot?
I need help with maths can someone help me plz.. is there a wats app group?
WY need
Fernando
How did you get $750? Laura Reply if y= 2x+sinx what is dy÷dx formon25 Reply does it teach you how to do algebra if you don't know how Kate Reply Liam borrowed a total of$35,000 to pay for college. He pays his parents 3% interest on the $8,000 he borrowed from them and pays the bank 6.8% on the rest. What average interest rate does he pay on the total$35,000? (Round your answer to the nearest tenth of a percent.)
exact definition of length by bilbao
the definition of length
literal meaning of length
francemichael
exact meaning of length
francemichael
exact meaning of length
francemichael
how many typos can we find...?
5
Joseph
In the LCM Prime Factors exercises, the LCM of 28 and 40 is 280. Not 420!
4x+7y=29,x+3y=11 substitute method of linear equation
substitute method of linear equation
Srinu
Solve one equation for one variable. Using the 2nd equation, x=11-3y. Substitute that for x in first equation. this will find y. then use the value for y to find the value for x.
bruce
I want to learn
Elizebeth
help
Elizebeth
I want to learn. Please teach me?
Wayne
1) Use any equation, and solve for any of the variables. Since the coefficient of x (the number in front of the x) in the second equation is 1 (it actually isn't shown, but 1 * x = x), use that equation. Subtract 3y from both sides (this isolates the x on the left side of the equal sign).
bruce
2) This results in x=11-3y. x is note in terms of y. Use that as the value of x and substitute for all x in the first equation. The first equation becomes 4(11-3y)+7y =29. Note that the only variable left in the first equation is the y. If you have multiple variable, then something is wrong.
bruce
3) Distribute (multiply) the 4 across 11-3y to get 44-12y. Add this to the 7y. So, the equation is now 44-5y=29.
bruce
4) Solve 44-5y=29 for y. Isolate the y by subtracting 44 from birth sides, resulting in -5y=-15. Now, divide birth sides by -5 (since you have -5y). This results in y=3. You now have the value of one variable.
bruce
5) The last step is to take the value of y from Step 4) and substitute into the 2nd equation. Therefore: x+3y=11 becomes x+3(3)=11. Then multiplying, x+9=11. Finally, solve for x by subtracting 9 from both sides. Therefore, x=2.
bruce
6) The ordered pair of (2, 3) is the proposed solution. To check, substitute those values into either equation. If the result is true, then the solution is correct. 4(2)+7(3)=8+21=29. TRUE! Finished.
bruce