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Multiply and Divide Integers

Multiply integers

Multiply Integers

Since multiplication is mathematical shorthand for repeated addition, our counter model can easily be applied to show multiplication of integers. Let’s look at this concrete model to see what patterns we notice. We will use the same examples that we used for addition and subtraction.

We remember that \(a\cdot b\) means add \(a,\ b\) times. Here, we are using the model shown in just to help us discover the pattern.

Now consider what it means to multiply \(5\) by \(-3.\) It means subtract \(5,3\) times. Looking at subtraction as taking away, it means to take away \(5,3\) times. But there is nothing to take away, so we start by adding neutral pairs as shown in .

In both cases, we started with \(\text{15}\) neutral pairs. In the case on the left, we took away \(\text{5},\text{3}\) times and the result was \(-\text{15}.\) To multiply \((-5)(-3),\) we took away \(-\text{5},\text{3}\) times and the result was \(\text{15}.\) So we found that

\[\begin{array}{lll}5\cdot 3=15 & & -5(3)=-15 \\ 5(-3)=-15 & & (-5)(-3)=15\end{array}\]

Notice that for multiplication of two signed numbers, when the signs are the same, the product is positive, and when the signs are different, the product is negative.

Example

Try it.

Multiply each of the following:

ⓐ \(\ -9\cdot 3\\)
ⓑ \(\ -2(-5)\\)
ⓒ \(\ 4(-8)\\)
ⓓ \(\ 7\cdot 6\)

Solution

ⓐ
	\(-9⋅3\)
Multiply, noting that the signs are different and so the product is negative.	\(-27\)

ⓑ
	\(-2(-5)\)
Multiply, noting that the signs are the same and so the product is positive.	\(10\)

ⓒ
	\(4(-8)\)
Multiply, noting that the signs are different and so the product is negative.	\(-32\)

ⓓ
	\(7⋅6\)
The signs are the same, so the product is positive.	\(42\)

When we multiply a number by \(1,\) the result is the same number. What happens when we multiply a number by\(-1?\) Let’s multiply a positive number and then a negative number by \(-1\) to see what we get.

\[\begin{array}{lll}-1\cdot 4 & \ & -1(-3) \\ -4 & \ & 3 \\ -4\ \text{is the opposite of}\ \text{4} & \ & \text{3}\ \text{is the opposite of}\ -3\end{array}\]

Each time we multiply a number by \(-1,\) we get its opposite.

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

Divide Integers

Division is the inverse operation of multiplication. So, \(15\div 3=5\) because \(5\cdot 3=15\) In words, this expression says that \(\text{15}\) can be divided into \(\text{3}\) groups of \(\text{5}\) each because adding five three times gives \(\text{15}.\) If we look at some examples of multiplying integers, we might figure out the rules for dividing integers.

\[\begin{array}{lllll}5\cdot 3=15\ \text{so}\ 15\div 3=5 & & & & -5(3)=-15\ \text{so}\ -15\div 3=-5 \\ (-5)(-3)=15\ \text{so}\ 15\div (-3)=-5 & & & & 5(-3)=-15\ \text{so}\ -15\div -3=5\end{array}\]

Division of signed numbers follows the same rules as multiplication. When the signs are the same, the quotient is positive, and when the signs are different, the quotient is negative.

Remember, you can always check the answer to a division problem by multiplying.

Example

Try it.

Divide each of the following:

ⓐ \(\ -27\div 3\\)
ⓑ \(\ -100\div (-4)\)

Solution

ⓐ
	\(-27\div 3\)
Divide, noting that the signs are different and so the quotient is negative.	\(-9\)

ⓑ
	\(-100\div (-4)\)
Divide, noting that the signs are the same and so the quotient is positive.	\(25\)

Just as we saw with multiplication, when we divide a number by \(1,\) the result is the same number. What happens when we divide a number by \(-1?\) Let’s divide a positive number and then a negative number by \(-1\) to see what we get.

\[\begin{array}{llll}8\div (-1) & & & -9\div (-1) \\ -8 & & & 9 \\ \text{-8 is the opposite of 8} & & & \text{9 is the opposite of -9}\end{array}\]

When we divide a number by, \(-1\) we get its opposite.

Example

Try it.

Divide each of the following:

ⓐ \(\ 16\div (-1)\\)
ⓑ \(\ -20\div (-1)\)

Solution

ⓐ
	\(16\div (-1)\)
The dividend, 16, is being divided by –1.	\(-16\)
Dividing a number by –1 gives its opposite.
Notice that the signs were different, so the result was negative.

ⓑ
	\(-20\div (-1)\)
The dividend, –20, is being divided by –1.	\(20\)
Dividing a number by –1 gives its opposite.

Notice that the signs were the same, so the quotient was positive.

Simplify Expressions with Integers

Now we’ll simplify expressions that use all four operations–addition, subtraction, multiplication, and division–with integers. Remember to follow the order of operations.

Example

Try it.

\(\text{Simplify:}\ 7(-2)+4(-7)-6.\)

Solution

We use the order of operations. Multiply first and then add and subtract from left to right.

	\(7(-2)+4(-7)-6\)
Multiply first.	\(-14+(-28)-6\)
Add.	\(-42-6\)
Subtract.	\(-48\)

Example

Try it.

Simplify:

ⓐ \(\ {(-2)}^{4}\\)
ⓑ \(\ {-2}^{4}\)

Solution

The exponent tells how many times to multiply the base.

ⓐ The exponent is \(4\) and the base is \(-2.\) We raise \(-2\) to the fourth power.

	\({(-2)}^{4}\)
Write in expanded form.	\((-2)(-2)(-2)(-2)\)
Multiply.	\(4(-2)(-2)\)
Multiply.	\(-8(-2)\)
Multiply.	\(16\)

ⓑ The exponent is \(4\) and the base is \(2.\) We raise \(2\) to the fourth power and then take the opposite.

	\(-{2}^{4}\)
Write in expanded form.	\(-(2⋅2⋅2⋅2)\)
Multiply.	\(-(4⋅2⋅2)\)
Multiply.	\(-(8⋅2)\)
Multiply.	\(-16\)

Example

Try it.

\(\text{Simplify:}\ 12-3(9-12).\)

Solution

According to the order of operations, we simplify inside parentheses first. Then we will multiply and finally we will subtract.

	\(12-3(9-12)\)
Subtract the parentheses first.	\(12-3(-3)\)
Multiply.	\(12-(-9)\)
Subtract.	\(21\)

Example

Try it.

Simplify: \(8(-9)\div {(-2)}^{3}.\)

Solution

We simplify the exponent first, then multiply and divide.

	\(8(-9)\div {(-2)}^{3}\)
Simplify the exponent.	\(8(-9)\div (-8)\)
Multiply.	\(-72\div (-8)\)
Divide.	\(9\)

Example

Try it.

\(\text{Simplify:}\ -30\div 2+(-3)(-7).\)

Solution

First we will multiply and divide from left to right. Then we will add.

	\(-30\div 2+(-3)(-7)\)
Divide.	\(-15+(-3)(-7)\)
Multiply.	\(-15+21\)
Add.	\(6\)

Evaluate Variable Expressions with Integers

Now we can evaluate expressions that include multiplication and division with integers. Remember that to evaluate an expression, substitute the numbers in place of the variables, and then simplify.

Example

Try it.

\(\text{Evaluate}\ 2{x}^{2}-3x+8\ \text{when}\ x=-4.\)

Solution



Simplify exponents.
Multiply.
Subtract.
Add.

Keep in mind that when we substitute \(-4\) for \(x,\) we use parentheses to show the multiplication. Without parentheses, it would look like \(2\cdot {-4}^{2}-3\cdot -4+8.\)

Example

Try it.

\(\text{Evaluate}\ 3x+4y-6\ \text{when}\ x=-1\ \text{and}\ y=2.\)

Solution


Substitute \(x=-1\) and \(y=2\).
Multiply.
Simplify.

Translate Word Phrases to Algebraic Expressions

Once again, all our prior work translating words to algebra transfers to phrases that include both multiplying and dividing integers. Remember that the key word for multiplication is product and for division is quotient.

Example

Try it.

Translate to an algebraic expression and simplify if possible: the product of \(-2\) and \(14.\)

Solution

The word product tells us to multiply.

	the product of \(-2\) and \(14\)
Translate.	\((-2)(14)\)
Simplify.	\(-28\)

Example

Try it.

Translate to an algebraic expression and simplify if possible: the quotient of \(-56\) and \(-7.\)

Solution

The word quotient tells us to divide.

	the quotient of −56 and −7
Translate.	\(-56\div (-7)\)
Simplify.	\(8\)

Key Concepts

Multiplication of Signed Numbers
- To determine the sign of the product of two signed numbers:
  Same Signs Product
  Two positives
  Two negatives Positive
  Positive
  
  Different Signs Product
  Positive • negative
  Negative • positive Negative
  Negative
Division of Signed Numbers
- To determine the sign of the quotient of two signed numbers:
  Same Signs Quotient
  Two positives
  Two negatives Positive
  Positive
  
  Different Signs Quotient
  Positive • negative
  Negative • Positive Negative
  Negative
Multiplication by \(-1\)
- Multiplying a number by \(-1\) gives its opposite: \(-1a=-a\)
Division by \(-1\)
- Dividing a number by \(-1\) gives its opposite: \(a\div (-1)=\text{-a}\)

Multiply and Divide Integers

Multiply Integers

In the following exercises, multiply each pair of integers.

Try it.

\(-4\cdot 8\)

Solution

−32

Try it.

\(-3\cdot 9\)

Try it.

\(-5(7)\)

Solution

−35

Try it.

\(-8(6)\)

Try it.

\(-18(-2)\)

Solution

Try it.

\(-10(-6)\)

Try it.

\(9(-7)\)

Solution

−63

Try it.

\(13(-5)\)

Try it.

\(-1\cdot 6\)

Solution

−6

Try it.

\(-1\cdot 3\)

Try it.

\(-1(-14)\)

Solution

Try it.

\(-1(-19)\)

Divide Integers

In the following exercises, divide.

Try it.

\(-24\div 6\)

Solution

−4

Try it.

\(-28\div 7\)

Try it.

\(56\div (-7)\)

Solution

−8

Try it.

\(35\div (-7)\)

Try it.

\(-52\div (-4)\)

Solution

Try it.

\(-84\div (-6)\)

Try it.

\(-180\div 15\)

Solution

−12

Try it.

\(-192\div 12\)

Try it.

\(49\div (-1)\)

Solution

−49

Try it.

\(62\div (-1)\)

Simplify Expressions with Integers

In the following exercises, simplify each expression.

Try it.

\(5(-6)+7(-2)-3\)

Solution

−47

Try it.

\(8(-4)+5(-4)-6\)

Try it.

\(-8(-2)-3(-9)\)

Solution

Try it.

\(-7(-4)-5(-3)\)

Try it.

\({(-5)}^{3}\)

Solution

−125

Try it.

\({(-4)}^{3}\)

Try it.

\({(-2)}^{6}\)

Solution

Try it.

\({(-3)}^{5}\)

Try it.

\(-{4}^{2}\)

Solution

−16

Try it.

\(-{6}^{2}\)

Try it.

\(-3(-5)(6)\)

Solution

Try it.

\(-4(-6)(3)\)

Try it.

\(-4\cdot 2\cdot 11\)

Solution

−88

Try it.

\(-5\cdot 3\cdot 10\)

Try it.

\((8-11)(9-12)\)

Solution

Try it.

\((6-11)(8-13)\)

Try it.

\(26-3(2-7)\)

Solution

Try it.

\(23-2(4-6)\)

Try it.

\(-10(-4)\div (-8)\)

Solution

−5

Try it.

\(-8(-6)\div (-4)\)

Try it.

\(65\div (-5)+(-28)\div (-7)\)

Solution

−9

Try it.

\(52\div (-4)+(-32)\div (-8)\)

Try it.

\(9-2[3-8(-2)]\)

Solution

−29

Try it.

\(11-3[7-4(-2)]\)

Try it.

\({(-3)}^{2}-24\div (8-2)\)

Solution

Try it.

\({(-4)}^{2}-32\div (12-4)\)

Evaluate Variable Expressions with Integers

In the following exercises, evaluate each expression.

Try it.

\(-2x+17\ \text{when}\\)

ⓐ \(\ x=8\\)
ⓑ \(\ x=-8\)

Solution

ⓐ 1
ⓑ 33

Try it.

\(-5y+14\ \text{when}\\)

ⓐ \(\ y=9\\)
ⓑ \(\ y=-9\)

Try it.

\(10-3m\ \text{when}\\)

ⓐ \(\ m=5\\)
ⓑ \(\ m=-5\)

Solution

ⓐ −5
ⓑ 25

Try it.

\(18-4n\ \text{when}\\)

ⓐ \(\ n=3\\)
ⓑ \(\ n=-3\)

Try it.

\({p}^{2}-5p+5\ \text{when}\ p=-1\)

Solution

Try it.

\({q}^{2}-2q+9\) when \(q=-2\)

Try it.

\(2{w}^{2}-3w+7\) when \(w=-2\)

Solution

Try it.

\(3{u}^{2}-4u+5\) when \(u=-3\)

Try it.

\(6x-5y+15\) when \(x=3\) and \(y=-1\)

Solution

Try it.

\(3p-2q+9\) when \(p=8\) and \(q=-2\)

Try it.

\(9a-2b-8\) when \(a=-6\) and \(b=-3\)

Solution

−56

Try it.

\(7m-4n-2\) when \(m=-4\) and \(n=-9\)

Translate Word Phrases to Algebraic Expressions

In the following exercises, translate to an algebraic expression and simplify if possible.

Try it.

The product of \(-3\) and 15

Solution

−3·15 = −45

Try it.

The product of \(-4\) and \(16\)

Try it.

The quotient of \(-60\) and \(-20\)

Solution

−60 ÷ (−20) = 3

Try it.

The quotient of \(-40\) and \(-20\)

Try it.

The quotient of \(-6\) and the sum of \(a\) and \(b\)

Solution

\(\frac{-6}{a+b}\)

Try it.

The quotient of \(-7\) and the sum of \(m\) and \(n\)

Try it.

The product of \(-10\) and the difference of \(p\ \text{and}\ q\)

Solution

−10 (p − q)

Try it.

The product of \(-13\) and the difference of \(c\ \text{and}\ d\)

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.

Translate the quotient of \(20\) and \(13\) into an algebraic expression.
If you missed this problem, review .

Revelar la respuesta

\(20\div 13\)
Add: \(-5+(-5)+(-5).\)
If you missed this problem, review .

Revelar la respuesta

\(-15\)
\(\text{Evaluate}\ n+4\ \text{when}\ n=-7.\)
If you missed this problem, review .

Revelar la respuesta

\(-3\)
Multiply each of the following:
1. ⓐ \(\ -9\cdot 3\\)
2. ⓑ \(\ -2(-5)\\)
3. ⓒ \(\ 4(-8)\\)
4. ⓓ \(\ 7\cdot 6\)
Revelar la respuesta

ⓐ
\(-9⋅3\)
Multiply, noting that the signs are different and so the product is negative. \(-27\)
ⓑ
\(-2(-5)\)
Multiply, noting that the signs are the same and so the product is positive. \(10\)
ⓒ
\(4(-8)\)
Multiply, noting that the signs are different and so the product is negative. \(-32\)

ⓓ
\(7⋅6\)
The signs are the same, so the product is positive. \(42\)
Multiply:
1. ⓐ \(\ -6\cdot 8\\)
2. ⓑ \(\ -4(-7)\\)
3. ⓒ \(\ 9(-7)\\)
4. ⓓ \(\ 5\cdot 12\)
Revelar la respuesta
1. ⓐ −48
2. ⓑ 28
3. ⓒ −63
4. ⓓ 60
Multiply:
1. ⓐ \(\ -8\cdot 7\\)
2. ⓑ \(\ -6(-9)\\)
3. ⓒ \(\ 7(-4)\\)
4. ⓓ \(\ 3\cdot 13\)
Revelar la respuesta
1. ⓐ −56
2. ⓑ 54
3. ⓒ −28
4. ⓓ 39
Multiply each of the following:
1. ⓐ \(\ -1\cdot 7\\)
2. ⓑ \(\ -1(-11)\)
Revelar la respuesta

ⓐ
The signs are different, so the product will be negative. \(-1⋅7\)
Notice that −7 is the opposite of 7. \(-7\)
ⓑ
The signs are the same, so the product will be positive. \(-1(-11)\)
Notice that 11 is the opposite of −11. \(11\)
Multiply.
1. ⓐ \(\ -1\cdot 9\\)
2. ⓑ \(\ -1\cdot (-17)\)
Revelar la respuesta
1. ⓐ −9
2. ⓑ 17
Multiply.
1. ⓐ \(\ -1\cdot 8\\)
2. ⓑ \(\ -1\cdot (-16)\)
Revelar la respuesta
1. ⓐ −8
2. ⓑ 16
Divide each of the following:
1. ⓐ \(\ -27\div 3\\)
2. ⓑ \(\ -100\div (-4)\)
Revelar la respuesta

ⓐ
\(-27\div 3\)
Divide, noting that the signs are different and so the quotient is negative. \(-9\)
ⓑ
\(-100\div (-4)\)
Divide, noting that the signs are the same and so the quotient is positive. \(25\)
Divide:
1. ⓐ \(\ -42\div 6\\)
2. ⓑ \(\ -117\div (-3)\)
Revelar la respuesta
1. ⓐ −7
2. ⓑ 39
Divide:
1. ⓐ \(\ -63\div 7\\)
2. ⓑ \(\ -115\div (-5)\)
Revelar la respuesta
1. ⓐ −9
2. ⓑ 23
Divide each of the following:
1. ⓐ \(\ 16\div (-1)\\)
2. ⓑ \(\ -20\div (-1)\)
Revelar la respuesta

ⓐ
\(16\div (-1)\)
The dividend, 16, is being divided by –1. \(-16\)
Dividing a number by –1 gives its opposite.
Notice that the signs were different, so the result was negative.
ⓑ
\(-20\div (-1)\)
The dividend, –20, is being divided by –1. \(20\)
Dividing a number by –1 gives its opposite.
Notice that the signs were the same, so the quotient was positive.
Divide:
1. ⓐ \(\ 6\div (-1)\\)
2. ⓑ \(\ -36\div (-1)\)
Revelar la respuesta
1. ⓐ −6
2. ⓑ 36
Divide:
1. ⓐ \(\ 28\div (-1)\\)
2. ⓑ \(\ -52\div (-1)\)
Revelar la respuesta
1. ⓐ −28
2. ⓑ 52
\(\text{Simplify:}\ 7(-2)+4(-7)-6.\)

Revelar la respuesta

We use the order of operations. Multiply first and then add and subtract from left to right.
\(7(-2)+4(-7)-6\)
Multiply first. \(-14+(-28)-6\)
Add. \(-42-6\)
Subtract. \(-48\)
Simplify:
\(8(-3)+5(-7)-4\)

Revelar la respuesta

−63
Simplify:
\(9(-3)+7(-8)-1\)

Revelar la respuesta

−84
Simplify:
1. ⓐ \(\ {(-2)}^{4}\\)
2. ⓑ \(\ {-2}^{4}\)
Revelar la respuesta

The exponent tells how many times to multiply the base.
ⓐ The exponent is \(4\) and the base is \(-2.\) We raise \(-2\) to the fourth power.
\({(-2)}^{4}\)
Write in expanded form. \((-2)(-2)(-2)(-2)\)
Multiply. \(4(-2)(-2)\)
Multiply. \(-8(-2)\)
Multiply. \(16\)
ⓑ The exponent is \(4\) and the base is \(2.\) We raise \(2\) to the fourth power and then take the opposite.
\(-{2}^{4}\)
Write in expanded form. \(-(2⋅2⋅2⋅2)\)
Multiply. \(-(4⋅2⋅2)\)
Multiply. \(-(8⋅2)\)
Multiply. \(-16\)
Simplify:
1. ⓐ \(\ {(-3)}^{4}\\)
2. ⓑ \(\ {-3}^{4}\)
Revelar la respuesta
1. ⓐ 81
2. ⓑ −81
Simplify:
1. ⓐ \(\ {(-7)}^{2}\\)
2. ⓑ \(\ -{7}^{2}\)
Revelar la respuesta
1. ⓐ 49
2. ⓑ −49
\(\text{Simplify:}\ 12-3(9-12).\)

Revelar la respuesta

According to the order of operations, we simplify inside parentheses first. Then we will multiply and finally we will subtract.
\(12-3(9-12)\)
Subtract the parentheses first. \(12-3(-3)\)
Multiply. \(12-(-9)\)
Subtract. \(21\)
Simplify:
\(17-4(8-11)\)

Revelar la respuesta

29
Simplify:
\(16-6(7-13)\)

Revelar la respuesta

52
Simplify: \(8(-9)\div {(-2)}^{3}.\)

Revelar la respuesta

We simplify the exponent first, then multiply and divide.
\(8(-9)\div {(-2)}^{3}\)
Simplify the exponent. \(8(-9)\div (-8)\)
Multiply. \(-72\div (-8)\)
Divide. \(9\)
Simplify:
\(12(-9)\div {(-3)}^{3}\)

Revelar la respuesta

4
Simplify:
\(18(-4)\div {(-2)}^{3}\)

Revelar la respuesta

9
\(\text{Simplify:}\ -30\div 2+(-3)(-7).\)

Revelar la respuesta

First we will multiply and divide from left to right. Then we will add.
\(-30\div 2+(-3)(-7)\)
Divide. \(-15+(-3)(-7)\)
Multiply. \(-15+21\)
Add. \(6\)
Simplify:
\(-27\div 3+(-5)(-6)\)

Revelar la respuesta

21
Simplify:
\(-32\div 4+(-2)(-7)\)

Revelar la respuesta

6
\(\text{Evaluate}\ 2{x}^{2}-3x+8\ \text{when}\ x=-4.\)

Revelar la respuesta

Simplify exponents.
Multiply.
Subtract.
Add.
Keep in mind that when we substitute \(-4\) for \(x,\) we use parentheses to show the multiplication. Without parentheses, it would look like \(2\cdot {-4}^{2}-3\cdot -4+8.\)
Evaluate:
\(3{x}^{2}-2x+6\ \text{when}\ x=-3\)

Revelar la respuesta

39
Evaluate:
\(4{x}^{2}-x-5\ \text{when}\ x=-2\)

Revelar la respuesta

13
\(\text{Evaluate}\ 3x+4y-6\ \text{when}\ x=-1\ \text{and}\ y=2.\)

Revelar la respuesta

Substitute \(x=-1\) and \(y=2\).
Multiply.
Simplify.
Evaluate:
\(7x+6y-12\ \text{when}\ x=-2\ \text{and}\ y=3\)

Revelar la respuesta

−8
Evaluate:
\(8x-6y+13\ \text{when}\ x=-3\ \text{and}\ y=-5\)

Revelar la respuesta

19
Translate to an algebraic expression and simplify if possible: the product of \(-2\) and \(14.\)

Revelar la respuesta

The word product tells us to multiply.
the product of \(-2\) and \(14\)
Translate. \((-2)(14)\)
Simplify. \(-28\)
Translate to an algebraic expression and simplify if possible:
\(\text{the product of -5 and 12}\)

Revelar la respuesta

−5 (12) = −60
Translate to an algebraic expression and simplify if possible:
\(\text{the product of 8 and -13}\)

Revelar la respuesta

8 (−13) = −104
Translate to an algebraic expression and simplify if possible: the quotient of \(-56\) and \(-7.\)

Revelar la respuesta

The word quotient tells us to divide.
the quotient of −56 and −7
Translate. \(-56\div (-7)\)
Simplify. \(8\)

Symbols used here

\pm

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

\neq

not equal
The two sides are different.

\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: Multiply and Divide Integers

Multiply integers
Divide integers
Simplify expressions with integers
Evaluate variable expressions with integers
Translate word phrases to algebraic expressions

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.

Prueba tu propio

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

Más en Arithmetic

Order of operations Fractions Percentages Powers and roots

Same signs	Product
•Two positives •Two negatives	Positive Positive

Different signs	Product
•Positive • negative •Negative • positive	Negative Negative

Same signs	Quotient
•Two positives •Two negatives	Positive Positive

Different signs	Quotient
•Positive & negative •Negative & positive	Negative Negative

Same Signs	Product
Two positives Two negatives	Positive Positive

Different Signs	Product
Positive • negative Negative • positive	Negative Negative

ⓐ
The signs are different, so the product will be negative.	\(-1⋅7\)
Notice that −7 is the opposite of 7.	\(-7\)

ⓑ
The signs are the same, so the product will be positive.	\(-1(-11)\)
Notice that 11 is the opposite of −11.	\(11\)