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Solve Equations with Decimals

Determine whether a decimal is a solution of an equation

Determine Whether a Decimal is a Solution of an Equation

Solving equations with decimals is important in our everyday lives because money is usually written with decimals. When applications involve money, such as shopping for yourself, making your family’s budget, or planning for the future of your business, you’ll be solving equations with decimals.

Now that we’ve worked with decimals, we are ready to find solutions to equations involving decimals. The steps we take to determine whether a number is a solution to an equation are the same whether the solution is a whole number, an integer, a fraction, or a decimal. We’ll list these steps here again for easy reference.

Example

Try it.

Determine whether each of the following is a solution of \(x-0.7=1.5\text{:}\)

ⓐ \(\ x=1\)ⓑ \(\ x=-0.8\)ⓒ \(\ x=2.2\)

Solution

ⓐ


Subtract.

Since \(x=1\) does not result in a true equation, \(1\) is not a solution to the equation.

ⓑ


Subtract.

Since \(x=-0.8\) does not result in a true equation, \(-0.8\) is not a solution to the equation.

ⓒ


Subtract.

Since \(x=2.2\) results in a true equation, \(2.2\) is a solution to the equation.

Solve Equations with Decimals

In previous chapters, we solved equations using the Properties of Equality. We will use these same properties to solve equations with decimals.

When you add, subtract, multiply or divide the same quantity from both sides of an equation, you still have equality.

Example

Try it.

Solve: \(y+2.3=-4.7.\)

Solution

We will use the Subtraction Property of Equality to isolate the variable.



Simplify.
Check:

Simplify.

Since \(y=-7\) makes \(y+2.3=-4.7\) a true statement, we know we have found a solution to this equation.

Example

Try it.

Solve: \(a-4.75=-1.39.\)

Solution

We will use the Addition Property of Equality.


Add 4.75 to each side, to undo the subtraction.
Simplify.
Check:

Since the result is a true statement, \(a=3.36\) is a solution to the equation.

Example

Try it.

Solve: \(-4.8=0.8n.\)

Solution

We will use the Division Property of Equality.

Use the Properties of Equality to find a value for \(n.\)


We must divide both sides by 0.8 to isolate n.
Simplify.
Check:

Since \(n=-6\) makes \(-4.8=0.8n\) a true statement, we know we have a solution.

Example

Try it.

Solve: \(\frac{p}{-1.8}=-6.5.\)

Solution

We will use the Multiplication Property of Equality.


Here, p is divided by −1.8. We must multiply by −1.8 to isolate p
Multiply.
Check:

A solution to \(\frac{p}{-1.8}=-6.5\) is \(p=11.7.\)

Translate to an Equation and Solve

Now that we have solved equations with decimals, we are ready to translate word sentences to equations and solve. Remember to look for words and phrases that indicate the operations to use.

Example

Try it.

Translate and solve: The difference of \(n\) and \(4.3\) is \(2.1.\)

Solution

Translate.
Add \(4.3\) to both sides of the equation.
Simplify.
Check:	Is the difference of \(n\) and 4.3 equal to 2.1?
Let \(n=6.4\):	Is the difference of 6.4 and 4.3 equal to 2.1?
Translate.
Simplify.

Example

Try it.

Translate and solve: The product of \(-3.1\) and \(x\) is \(5.27.\)

Solution

Translate.
Divide both sides by \(-3.1\).
Simplify.
Check:	Is the product of −3.1 and \(x\) equal to \(5.27\)?
Let \(x=-1.7\):	Is the product of \(-3.1\) and \(-1.7\) equal to \(5.27\)?
Translate.
Simplify.

Example

Try it.

Translate and solve: The quotient of \(p\) and \(-2.4\) is \(6.5.\)

Solution

Translate.
Multiply both sides by \(-2.4\).
Simplify.
Check:	Is the quotient of \(p\) and \(-2.4\) equal to \(6.5\)?
Let \(p=-15.6:\)	Is the quotient of \(-15.6\) and \(-2.4\) equal to \(6.5\)?
Translate.
Simplify.

Example

Try it.

Translate and solve: The sum of \(n\) and \(2.9\) is \(1.7.\)

Solution

Translate.
Subtract \(2.9\) from each side.
Simplify.
Check:	Is the sum \(n\) and \(2.9\) equal to \(1.7\)?
Let \(n=-1.2:\)	Is the sum \(-1.2\) and \(2.9\) equal to \(1.7\)?
Translate.
Simplify.

Key Concepts

Determine whether a number is a solution to an equation.
- Substitute the number for the variable in the equation.
- Simplify the expressions on both sides of the equation.
- Determine whether the resulting equation is true.
  If so, the number is a solution.
  If not, the number is not a solution.
Properties of Equality

Subtraction Property of Equality	Addition Property of Equality
For any numbers \(a\), \(b\), and \(c\), \(\begin{array}{llll}\text{If} & a & = & b \\ \text{then} & a-c & = & b-c\end{array}\)	For any numbers \(a\), \(b\), and \(c\), \(\begin{array}{llll}\text{If} & a & = & b \\ \text{then} & a+c & = & b+c\end{array}\)
Division of Property of Equality	Multiplication Property of Equality
For any numbers \(a\), \(b\), and \(c\ne 0\), \(\begin{array}{llll}\text{If} & a & = & b \\ \text{then} & \frac{a}{c} & = & \frac{b}{c}\end{array}\)	For any numbers \(a\), \(b\), and \(c\), \(\begin{array}{llll}\text{If} & a & = & b \\ \text{then} & a⋅c & = & b⋅c\end{array}\)

Solve Equations with Decimals

Determine Whether a Decimal is a Solution of an Equation

In the following exercises, determine whether each number is a solution of the given equation.

Try it.

\(x-0.8=2.3\)
ⓐ \(\ x=2\)ⓑ \(\ x=-1.5\)ⓒ \(\ x=3.1\)

Solution

ⓐ no
ⓑ no
ⓒ yes

Try it.

\(y+0.6=-3.4\)
ⓐ \(\ y=-4\)ⓑ \(\ y=-2.8\)ⓒ \(\ y=2.6\)

Try it.

\(\frac{h}{1.5}=-4.3\)
ⓐ \(\ h=6.45\)ⓑ \(\ h=-6.45\)ⓒ \(\ h=-2.1\)

Solution

ⓐ no
ⓑ yes
ⓒ no

Try it.

\(0.75k=-3.6\)
ⓐ \(\ k=-0.48\)ⓑ \(\ k=-4.8\)ⓒ \(\ k=-2.7\)

Solve Equations with Decimals

In the following exercises, solve the equation.

Try it.

\(y+2.9=5.7\)

Solution

y = 2.8

Try it.

\(m+4.6=6.5\)

Try it.

\(f+3.45=2.6\)

Solution

f = −0.85

Try it.

\(h+4.37=3.5\)

Try it.

\(a+6.2=-1.7\)

Solution

a = −7.9

Try it.

\(b+5.8=-2.3\)

Try it.

\(c+1.15=-3.5\)

Solution

c = −4.65

Try it.

\(d+2.35=-4.8\)

Try it.

\(n-2.6=1.8\)

Solution

n = 4.4

Try it.

\(p-3.6=1.7\)

Try it.

\(x-0.4=-3.9\)

Solution

x = −3.5

Try it.

\(y-0.6=-4.5\)

Try it.

\(j-1.82=-6.5\)

Solution

j = −4.68

Try it.

\(k-3.19=-4.6\)

Try it.

\(m-0.25=-1.67\)

Solution

m = −1.42

Try it.

\(q-0.47=-1.53\)

Try it.

\(0.5x=3.5\)

Solution

x = 7

Try it.

\(0.4p=9.2\)

Try it.

\(-1.7c=8.5\)

Solution

c = −5

Try it.

\(-2.9x=5.8\)

Try it.

\(-1.4p=-4.2\)

Solution

p = 3

Try it.

\(-2.8m=-8.4\)

Try it.

\(-120=1.5q\)

Solution

q = −80

Try it.

\(-75=1.5y\)

Try it.

\(0.24x=4.8\)

Solution

x = 20

Try it.

\(0.18n=5.4\)

Try it.

\(-3.4z=-9.18\)

Solution

z = 2.7

Try it.

\(-2.7u=-9.72\)

Try it.

\(\frac{a}{0.4}=-20\)

Solution

a = −8

Try it.

\(\frac{b}{0.3}=-9\)

Try it.

\(\frac{x}{0.7}=-0.4\)

Solution

x = −0.28

Try it.

\(\frac{y}{0.8}=-0.7\)

Try it.

\(\frac{p}{-5}=-1.65\)

Solution

p = 8.25

Try it.

\(\frac{q}{-4}=-5.92\)

Try it.

\(\frac{r}{-1.2}=-6\)

Solution

r = 7.2

Try it.

\(\frac{s}{-1.5}=-3\)

Mixed Practice

In the following exercises, solve the equation. Then check your solution.

Try it.

\(x-5=-11\)

Solution

x = −6

Try it.

\(-\frac{2}{5}=x+\frac{3}{4}\)

Try it.

\(p+8=-2\)

Solution

p = −10

Try it.

\(p+\frac{2}{3}=\frac{1}{12}\)

Try it.

\(-4.2m=-33.6\)

Solution

m = 8

Try it.

\(q+9.5=-14\)

Try it.

\(q+\frac{5}{6}=\frac{1}{12}\)

Solution

\(q=-\frac{3}{4}\)

Try it.

\(\frac{8.6}{15}=-d\)

Try it.

\(\frac{7}{8}m=\frac{1}{10}\)

Solution

\(m=\frac{4}{35}\)

Try it.

\(\frac{j}{-6.2}=-3\)

Try it.

\(-\frac{2}{3}=y+\frac{3}{8}\)

Solution

\(y=-\frac{25}{24}\)

Try it.

\(s-1.75=-3.2\)

Try it.

\(\frac{11}{20}=-f\)

Solution

\(f=-\frac{11}{20}\)

Try it.

\(-3.6b=2.52\)

Try it.

\(-4.2a=3.36\)

Solution

a = −0.8

Try it.

\(-9.1n=-63.7\)

Try it.

\(r-1.25=-2.7\)

Solution

r = −1.45

Try it.

\(\frac{1}{4}n=\frac{7}{10}\)

Try it.

\(\frac{h}{-3}=-8\)

Solution

h = 24

Try it.

\(y-7.82=-16\)

Translate to an Equation and Solve

In the following exercises, translate and solve.

Try it.

The difference of \(n\) and \(1.9\) is \(3.4.\)

Solution

\(n-1.9=3.4;5.3\)

Try it.

The difference \(n\) and \(1.5\) is \(0.8.\)

Try it.

The product of \(-6.2\) and \(x\) is \(-4.96.\)

Solution

−6.2x = −4.96; 0.8

Try it.

The product of \(-4.6\) and \(x\) is \(-3.22.\)

Try it.

The quotient of \(y\) and \(-1.7\) is \(-5.\)

Solution

\(\frac{y}{-1.7}=-5;\ 8.5\)

Try it.

The quotient of \(z\) and \(-3.6\) is \(3.\)

Try it.

The sum of \(n\) and \(-7.3\) is \(2.4.\)

Solution

n + (−7.3) = 2.4; 9.7

Try it.

The sum of \(n\) and \(-5.1\) is \(3.8.\)

Condensed — the full section is in OpenStax Prealgebra 2e.

Practice (40)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

Evaluate \(x+\frac{2}{3}\ \text{when}\ x=-\frac{1}{4}.\)
If you missed this problem, review .

Lafunua jibu

\(\frac{5}{12}\)
Evaluate \(15-y\) when \(y=-5.\)
If you missed this problem, review .

Lafunua jibu

\(20\)
Solve \(\frac{n}{-7}=42.\)
If you missed this problem, review .

Lafunua jibu

\(-294\)
Determine whether each of the following is a solution of \(x-0.7=1.5\text{:}\)
ⓐ \(\ x=1\)ⓑ \(\ x=-0.8\)ⓒ \(\ x=2.2\)

Lafunua jibu

ⓐ
Subtract.
Since \(x=1\) does not result in a true equation, \(1\) is not a solution to the equation.
ⓑ
Subtract.
Since \(x=-0.8\) does not result in a true equation, \(-0.8\) is not a solution to the equation.
ⓒ
Subtract.
Since \(x=2.2\) results in a true equation, \(2.2\) is a solution to the equation.
Determine whether each value is a solution of the given equation.
\(x-0.6=1.3:\\)ⓐ \(\ x=0.7\)ⓑ \(\ x=1.9\)ⓒ \(\ x=-0.7\)
Lafunua jibu
1. ⓐ no
2. ⓑ yes
3. ⓒ no
Determine whether each value is a solution of the given equation.
\(y-0.4=1.7:\\)ⓐ \(\ y=2.1\)ⓑ \(\ y=1.3\)ⓒ \(\ -1.3\)
Lafunua jibu
1. ⓐ yes
2. ⓑ no
3. ⓒ no
Solve: \(y+2.3=-4.7.\)

Lafunua jibu

We will use the Subtraction Property of Equality to isolate the variable.

Simplify.
Check:
Simplify.
Since \(y=-7\) makes \(y+2.3=-4.7\) a true statement, we know we have found a solution to this equation.
Solve: \(y+2.7=-5.3.\)

Lafunua jibu

y = −8
Solve: \(y+3.6=-4.8.\)

Lafunua jibu

y = −8.4
Solve: \(a-4.75=-1.39.\)

Lafunua jibu

We will use the Addition Property of Equality.

Add 4.75 to each side, to undo the subtraction.
Simplify.
Check:

Since the result is a true statement, \(a=3.36\) is a solution to the equation.
Solve: \(a-3.93=-2.86.\)

Lafunua jibu

a = 1.07
Solve: \(n-3.47=-2.64.\)

Lafunua jibu

n = 0.83
Solve: \(-4.8=0.8n.\)

Lafunua jibu

We will use the Division Property of Equality.
Use the Properties of Equality to find a value for \(n.\)

We must divide both sides by 0.8 to isolate n.
Simplify.
Check:

Since \(n=-6\) makes \(-4.8=0.8n\) a true statement, we know we have a solution.
Solve: \(-8.4=0.7b.\)

Lafunua jibu

b = −12
Solve: \(-5.6=0.7c.\)

Lafunua jibu

c = −8
Solve: \(\frac{p}{-1.8}=-6.5.\)

Lafunua jibu

We will use the Multiplication Property of Equality.

Here, p is divided by −1.8. We must multiply by −1.8 to isolate p
Multiply.
Check:
A solution to \(\frac{p}{-1.8}=-6.5\) is \(p=11.7.\)
Solve: \(\frac{c}{-2.6}=-4.5.\)

Lafunua jibu

c = 11.7
Solve: \(\frac{b}{-1.2}=-5.4.\)

Lafunua jibu

b = 6.48
Translate and solve: The difference of \(n\) and \(4.3\) is \(2.1.\)

Lafunua jibu

Translate.
Add \(4.3\) to both sides of the equation.
Simplify.
Check: Is the difference of \(n\) and 4.3 equal to 2.1?
Let \(n=6.4\): Is the difference of 6.4 and 4.3 equal to 2.1?
Translate.
Simplify.
Translate and solve: The difference of \(y\) and \(4.9\) is \(2.8.\)

Lafunua jibu

y − 4.9 = 2.8; y = 7.7
Translate and solve: The difference of \(z\) and \(5.7\) is \(3.4.\)

Lafunua jibu

z − 5.7 = 3.4; z = 9.1
Translate and solve: The product of \(-3.1\) and \(x\) is \(5.27.\)

Lafunua jibu

Translate.
Divide both sides by \(-3.1\).
Simplify.
Check: Is the product of −3.1 and \(x\) equal to \(5.27\)?
Let \(x=-1.7\): Is the product of \(-3.1\) and \(-1.7\) equal to \(5.27\)?
Translate.
Simplify.
Translate and solve: The product of \(-4.3\) and \(x\) is \(12.04.\)

Lafunua jibu

−4.3x = 12.04; x = −2.8
Translate and solve: The product of \(-3.1\) and \(m\) is \(26.66.\)

Lafunua jibu

−3.1m = 26.66; m = −8.6

Translate and solve: The quotient of \(p\) and \(-2.4\) is \(6.5.\)

Lafunua jibu

Translate.
Multiply both sides by \(-2.4\).
Simplify.
Check:	Is the quotient of \(p\) and \(-2.4\) equal to \(6.5\)?
Let \(p=-15.6:\)	Is the quotient of \(-15.6\) and \(-2.4\) equal to \(6.5\)?
Translate.
Simplify.

Translate and solve: The quotient of \(q\) and \(-3.4\) is \(4.5.\)

Lafunua jibu

\(\frac{q}{-3.4}=4.5;\ q=-15.3\)
Translate and solve: The quotient of \(r\) and \(-2.6\) is \(2.5.\)

Lafunua jibu

\(\frac{r}{-2.6}=2.5;\ r=-6.5\)
Translate and solve: The sum of \(n\) and \(2.9\) is \(1.7.\)

Lafunua jibu

Translate.
Subtract \(2.9\) from each side.
Simplify.
Check: Is the sum \(n\) and \(2.9\) equal to \(1.7\)?
Let \(n=-1.2:\) Is the sum \(-1.2\) and \(2.9\) equal to \(1.7\)?
Translate.
Simplify.
Translate and solve: The sum of \(j\) and \(3.8\) is \(2.6.\)

Lafunua jibu

j + 3.8 = 2.6; j = −1.2
Translate and solve: The sum of \(k\) and \(4.7\) is \(0.3.\)

Lafunua jibu

k + 4.7 = 0.3; k = −4.4
\(y+2.9=5.7\)

Lafunua jibu

y = 2.8
\(m+4.6=6.5\)
\(f+3.45=2.6\)

Lafunua jibu

f = −0.85
\(h+4.37=3.5\)
\(a+6.2=-1.7\)

Lafunua jibu

a = −7.9
\(b+5.8=-2.3\)

Symbols used here

\neq

not equal
The two sides are different.

\pm

plus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.

\leq,\ \geq

less/greater than or equal
Inequalities that allow equality; < and > exclude it.

\approx

approximately equal
Equal to the precision shown, not exactly.

\sqrt{x},\ \sqrt[n]{x}

square root, n-th root
The non-negative number whose square (n-th power) is x.

a \bmod n

remainder
What is left after dividing a by n.

per cent
Per hundred: 15% = 15/100.

a : b,\ \frac{a}{b}

ratio, fraction
a for every b; a divided by b.

How to: Solve Equations with Decimals

Determine whether a decimal is a solution of an equation
Solve equations with decimals
Translate to an equation and solve
Substitute the number for the variable in the equation.
Simplify the expressions on both sides of the equation.
Determine whether the resulting equation is true.
If so, the number is a solution.
If not, the number is not a solution.

Questions people ask

Why does the order of operations matter?

Because 2 + 3 × 4 would otherwise be two different numbers. The convention (brackets, exponents, multiplication and division, addition and subtraction) exists so every reader gets the same value from the same expression.

How do I check an arithmetic answer?

Estimate first (round every number and compute roughly), then compare. If the estimate and the exact answer disagree by more than a little, one of them is wrong. The solver shows every operation, so you can find which line went astray.

Why are fractions harder than decimals?

They are not harder, they are more exact: 1/3 is a precise number, 0.333 is an approximation. Fractions need a common denominator to add, which is the one extra step people trip on.

Jaribu kufanya mambo yako mwenyewe

Parts of this page are adapted from OpenStax Prealgebra 2e (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.

Mengi zaidi katika Arithmetic

Order of operations Fractions Percentages Powers and roots

Subtraction Property of Equality For any numbers \(a,b,\ \text{and}\ c,\) If \(a=b,\) then \(a-c=b-c.\)	Addition Property of Equality For any numbers \(a,b,\ \text{and}\ c,\) If \(a=b,\) then \(a+c=b+c.\)
The Division Property of Equality For any numbers \(a,b,\ \text{and}\ c,\ \text{and}\ c\ne 0\) If \(a=b,\) then \(\frac{a}{c}=\frac{b}{c}\)	The Multiplication Property of Equality For any numbers \(a,b,\ \text{and}\ c,\) If \(a=b,\) then \(ac=bc\)