Integers: Calculating with Negative Numbers

A minus sign gives a number a place on the left of zero. It can also give that number a meaning, such as a height below sea level.

Integers and rational numbers

Integers are the whole numbers and their opposites:

Positive numbers are greater than zero. Negative numbers are less than zero. Zero is neither positive nor negative. You can leave out a positive sign: $+5$ and $5$ name the same number. Positive and negative numbers are also called signed numbers or directed numbers.

A rational number can be written as a fraction of integers, with a denominator other than zero. Integers are rational too: $-7=\frac{-7}{1}$. Fractions and decimals can sit between the integers. For example, $-2.5=-\frac{5}{2}$ is halfway between $-3$ and $-2$.

Compare positions

Use equal spacing for equal steps on a number line. Numbers get greater as you move right. The same rule works on both sides of zero:

The symbol $<$ means “is less than.” The symbol $>$ means “is greater than.” Although $8$ is greater than $3$, negative eight is farther left than negative three. A temperature of $-8$ °C is colder than $-3$ °C.

To compare numbers in different forms, first write them in one form. Consider $-0.6$, $-\frac{1}{2}$, and $-0.65$.

1. Change the fraction: $-\frac{1}{2}=-0.5$.
2. Write matching decimal places: $-0.60$, $-0.50$, and $-0.65$.
3. Put the farthest-left value first: $-0.65<-0.60<-0.50$.

Adding a final zero after a decimal does not change its value. It helps show that $0.60$ is sixty hundredths. On the negative side, sixty-five hundredths below zero is lower than sixty hundredths below zero.

Give zero a meaning

At sea level, elevation is zero. An elevation of $-40$ meters means 40 meters below sea level. On a Fahrenheit thermometer, $-12$ °F means 12 degrees below zero on that scale. An account balance of −$30 means the account owes $30.

Directions can use signs too. If a gain of 5 yards is $+5$, a loss of 5 yards is $-5$. If the ground floor is zero, two floors below it can be labeled $-2$. Keep the same choice of positive direction throughout a problem.

A point at $-\frac{1}{4}$ lies between $-1$ and $0$. Split that interval into four equal parts. The first quarter-step left from zero marks the point. The minus sign locates the point; it does not change the size of each quarter-step.

Two numbers can be equally far from zero while lying on different sides. This gives two useful ideas: opposites and absolute value.

Find an opposite

The opposite of a number lies the same distance from zero on the other side. The opposite of $6$ is $-6$. The opposite of $-2.5$ is $2.5$. Zero is its own opposite.

A number and its opposite add to zero:

This is why an opposite is also called an additive inverse. Opposite charges give a real example. A hydrogen atom has a proton with charge $+1$ and an electron with charge $-1$. Their total charge is zero.

The notation $-a$ means “the opposite of $a$.” Its value depends on $a$. If $a=-6$, then $-a=6$. Taking the opposite twice returns the starting number: $-(-4)=4$.

Show cancellation with counters

Let a positive counter stand for $+1$ and a negative counter for $-1$. One of each makes a zero pair. You can add or remove a zero pair without changing the total.

Suppose you have 3 positive counters and 5 negative counters. Pair each positive with a negative. Three pairs cancel, leaving 2 negative counters. The total is $-2$. There are eight counters, but their signed total is not eight.

Measure size with absolute value

Absolute value is distance from zero. Vertical bars name this distance:

Distance cannot be negative. Opposites have the same absolute value because they are equally far from zero. Absolute value does not mean “change the sign”: the absolute value of a positive number stays positive.

A minus sign outside the bars acts after the distance is found:

First, the bars give $5$. Then the outside minus sign takes its opposite.

Separate size from order

A balance of −$45 is less than a balance of −$20. But the first debt is larger in size: 45 dollars compared with 20 dollars. A negative value can be lower while having a greater absolute value.

To find the distance between two numbers, subtract and take the absolute value. Between $-3$ and $5$:

Reversing the subtraction gives $\mathrm{\mid }-3-5\mathrm{\mid }=\mathrm{\mid }-8\mathrm{\mid }=8$. The direction changes, but the distance does not. You can check by counting 3 steps to zero and 5 more to the other point.

To add on a number line, start at the first number. The number you add tells you the direction and size of the move.

Move right or left

Adding a positive number moves right. Adding a negative number moves left:

For the first sum, move 5 steps right from $-2$. You reach zero after 2 steps, then continue 3 more. For the second, move 6 left from $4$. Passing zero does not change the direction of the move.

Add numbers with the same sign

Add their absolute values and keep their shared sign:

Three negative counters and eight negative counters make eleven negative counters. There are no positive counters to cancel them. Two debts also combine into a larger debt: owing $3 and then another $8 means owing $11.

This is why “two negatives make a positive” does not work for addition. Two negative amounts add to a negative amount.

Add numbers with different signs

Positive and negative amounts cancel each other. Subtract the smaller absolute value from the larger one. Keep the sign of the number with the larger absolute value.

For $-9+4$:

1. Compare sizes: $9>4$, so the negative part is larger.
2. Find what is left after cancellation: $9-4=5$.
3. Keep the negative sign: $-9+4=-5$.

With counters, four positive counters cancel four of the nine negatives. Five negatives remain. If the positive part were larger, the result would be positive: $-7+12=5$.

Equal opposite amounts cancel completely, so $-7+7=0$. Adding zero makes no move: $-4+0=-4$.

Group a longer sum

The commutative property lets you change the order in addition. The associative property lets you change the grouping. Keep each number’s sign attached to it.

For $-7+15+(-3)+8$, group the positives and the negatives:

Then combine them: $23+(-10)=13$. You can also group a number with its opposite when both appear in a sum.

Check the direction of a change

At 7 a.m., a thermometer reads $-6$ °F. It rises 15 degrees. The new temperature is $-6+15=9$ °F. The rise moves the reading right, even though it starts below zero.

Before calculating $-38+21$, notice that 38 is greater than 21. The negative amount will not cancel completely. The answer must be negative; its size is $38-21=17$, giving $-17$.

Subtraction has one rule that works for every signed number: add the opposite of the number being subtracted.

Keep the first number

For $5-8$, add negative eight: $5+(-8)=-3$. For $-4-6$, start at negative four and move six left: $-4+(-6)=-10$.

When the second number is negative, its opposite is positive:

The first number stays $-3$. Only the subtraction and the sign of the second number change. Changing both numbers would give a different calculation.

Parentheses make the two jobs of a minus sign clear. In $6-(-2)$, the first minus means subtract. The second belongs to negative two. Read it as “six minus negative two.” Adding the opposite gives $6+2=8$.

Why removing negatives increases the total

Use counters for $2-(-7)$. Start with 2 positive counters. You need to remove 7 negative counters, but there are none yet.

1. Add 7 zero pairs. The total is still 2.
2. You now have 9 positive and 7 negative counters.
3. Remove the 7 negative counters.

Nine positive counters remain. So $2-(-7)=9$. Adding the zero pairs made the removal possible without changing the starting value.

Find a change

A signed change is final value minus starting value. If the temperature goes from $-8$ °C to $3$ °C, the change is

That is a rise of 11 degrees. On the number line, it is the move from $-8$ to $3$. If the temperature goes from $4$ °C to $-5$ °C, the change is $-5-4=-9$: a drop of 9 degrees.

Keep the order when subtracting. For example, $3-8=-5$ but $8-3=5$. Reversing a subtraction gives the opposite result. Distance uses the absolute value of that result, so it has no negative direction.

Handle several signs

Rewrite every subtraction before regrouping a longer calculation:

Now the positive total is $14$ and the negative total is $-9$. Together they give $5$. Working left to right also gives $7$, then $11$, then $5$.

The shortcut “minus a negative becomes plus” applies to the signs beside each other in $5-(-3)$. It does not make $-5-3$ positive. That calculation moves 3 left from $-5$ and ends at $-8$.

A story tells you what the numbers and signs mean. First choose the starting value, then decide whether each change raises or lowers it.

Write the change with its sign

Use this relationship:

A diver starts 8 meters below the surface. Take the surface as zero and upward as positive. Going down another 14 meters is a change of $-14$ meters:

The diver is now 22 meters below the surface. A negative answer here gives a position, not a distance traveled.

Words such as “rises,” “gain,” and “deposit” often describe positive changes. “Falls,” “loss,” and “withdrawal” often describe negative changes. Read the whole sentence: “How much did it fall?” asks for the size of a drop, while “What was the change?” can ask for a signed value.

Follow a money balance

An account holds $40. A $65 purchase leaves a balance of

The account is overdrawn by $25. A later $100 deposit gives $-25+100=75$, so the balance is now $75. Crossing zero means the deposit first clears the debt, then leaves money in the account.

Compare a high and a low

To find a range, subtract the low value from the high value. With a high of $38$ °C and a low of $-27$ °C:

The range is 65 degrees. From $-27$ to zero takes 27 degrees, then reaching 38 takes 38 more. Calculating $38-27$ would lose the part below zero.

Signed scores can be added in the same way. Golf scores of $-3$, $+1$, $-2$, and $0$ relative to par add to $-4$. The combined score is 4 under par.

Work back to a starting value

If you know the end and the change, subtract the change:

A temperature fell 6 degrees and is now $-4$ °C. The change was $-6$, so the starting temperature was

Check by following the story forward: $2+(-6)=-4$.

Explain the answer

A submarine at $-120$ meters rises 45 meters. Its final position is $-120+45=-75$ meters, or 75 meters below sea level. The final value is greater than the starting value, as a rise requires. Saying “75 meters below sea level” keeps both the unit and the direction clear.

Multiplication can combine equal changes. Four losses of 6 points give $4\times (-6)=-24$ points. A positive whole-number multiplier counts how many equal amounts to add.

From repeated addition to a sign rule

Multiplication allows you to swap the factors, so $(-4)\times 3=-12$ too. A positive times a negative gives a negative product.

To see why two negatives give a positive, follow a pattern:

Each time the second factor decreases by 1, the product increases by 3. Continuing the pattern through zero gives $(-3)(-2)=6$.

Explain it with distribution

The distributive property says that multiplying a sum gives the same result as multiplying each part and adding the products. Because $2+(-2)=0$:

Multiplying each part gives $-6+(-3)(-2)=0$. The missing product must be $6$ to cancel $-6$. A positive product keeps the distributive property working for negative factors.

Find the sign, then the size

For two nonzero factors:

• Same signs give a positive product: $(-7)(-8)=56$.
• Different signs give a negative product: $(-6)(9)=-54$.

Multiply the absolute values to find the size. The larger factor does not decide the sign. For example, both $(-9)(2)$ and $9(-2)$ equal $-18$.

Any number multiplied by zero gives zero. Multiplying by $-1$ gives the opposite: $(-1)(-8)=8$.

Parentheses next to each other mean multiplication. So $(-4)(5)$, $-4\times 5$, and $-4\cdot 5$ all mean $-20$. By contrast, $-4-5$ subtracts 5 and gives $-9$.

Use the properties to make work easier

Distribute a negative factor to both parts:

The second product is positive because $-4$ multiplies $-3$. You can also reorder and regroup factors: $(-25)\times 7\times (-4)=100\times 7=700$.

A signed time can describe an earlier moment. If a temperature falls steadily by 3 degrees per hour, its rate is $-3$. Two hours ago has time $-2$ relative to now. The change from now to that earlier reading is $(-3)(-2)=6$ degrees, so it was warmer then.

Division asks what factor is missing. To find $12÷(-3)$, ask, “What number times negative three gives twelve?”

Use the multiplication signs

The sign rules for division match multiplication. With nonzero numbers, same signs give a positive quotient. Different signs give a negative quotient:

Divide the absolute values to find the size. Then attach the sign. Check by multiplying the quotient by the original divisor. For $-56÷8=-7$, the check is $(-7)\times 8=-56$.

Order matters. The quotient $-12÷3=-4$ differs from $3÷(-12)=-\frac{1}{4}$. Swapping the numbers changes which factor is missing.

Know where zero can go

Zero divided by a nonzero number is zero. For example, $0÷(-5)=0$ because $0\times (-5)=0$.

Division by zero is undefined. No number multiplied by zero can give $-8$, so $-8÷0$ has no answer. For $0÷0$, every number times zero gives zero. That cannot choose one quotient either.

Read a fraction bar as division

A fraction bar separates the dividend above from the divisor below:

One negative sign can be written in front, on top, or on the bottom:

Two negative signs give a positive value: $\frac{-3}{-4}=\frac{3}{4}$. The denominator must still be nonzero.

Dividing integers need not give an integer. For example,

It is a rational number because it can be written as a fraction of integers with a nonzero denominator.

Interpret equal shares and rates

Four people share a debt of $84 equally. Each signed share is $-84÷4=-21$ dollars, so each person owes $21. Dividing the debt does not turn it into money they own.

If the temperature drops 18 degrees over 6 hours, the signed total change is $-18$ degrees. The average change per hour is $-18÷6=-3$ degrees per hour. Multiplying this rate by 6 hours returns the total change.

Several negative factors can look confusing. Pairing negatives lets you decide the sign before multiplying their sizes.

Count negative factors

Each pair of negative factors gives a positive product. If no factor is zero:

• An even number of negative factors gives a positive product.
• An odd number leaves one unpaired negative factor, giving a negative product.

For $(-2)(-5)(-3)(4)$, count three negative factors. The product is negative. Multiply sizes in stages: $2\times 5=10$, $10\times 3=30$, and $30\times 4=120$. The answer is $-120$.

Positive factors change the size, but not the sign. Check for zero before counting: $(-4)\times 0\times (-7)\times (-9)=0$.

A chain of multiplication and division is worked from left to right:

The three negative numbers predict a negative result. The left-to-right work finds its size correctly.

A power counts equal factors

A base is the number being raised to a power. A positive whole-number exponent tells how many equal factors to use:

The exponent 3 gives three factors, not three times the base. Calculating $3\times (-2)$ would give $-6$ and answer a different question.

For a negative base, an even positive whole-number exponent gives a positive value. An odd one gives a negative value. Thus $(-2{)}^4=16$, $(-1{)}^{10}=1$, and $(-1{)}^7=-1$.

Read the parentheses before the exponent

Compare:

Parentheses make the negative sign part of the base. Without them, the exponent acts before the minus sign. In the same way, $(-2{)}^4=16$ but $-2^4=-16$.

Squares and square roots

The square of a nonzero number is positive. Both $5^2$ and $(-5{)}^2$ equal $25$ because each multiplies two factors with the same sign. Zero squared is zero.

The symbol $\sqrt{25}$ names the positive square root, $5$. Although $-5$ also has square $25$, it is not the value of that symbol. No real number has square $-25$, since a real square is never negative.

A fraction, a mixed number, and a decimal can name the same point on a number line. Changing the form keeps the value and its sign.

Keep the sign with the whole value

These forms all name negative three fourths:

The sign is usually easiest to read in front. Two negative signs, one above and one below the fraction bar, would give a positive fraction.

For a negative mixed number, the minus sign applies to the whole amount:

Think of two whole units and another third, all left of zero. Reading it as $-2+\frac{1}{3}$ gives $-\frac{5}{3}$, which lies on the other side of $-2$. That is a different point.

To convert a mixed number, multiply the whole-number size by the denominator, add the numerator, and keep the sign. This makes the entire amount negative.

Convert a fraction to a decimal

Divide the numerator by the denominator:

This decimal terminates, or ends. For $-\frac{2}{3}$, division gives $-0.666\dots$. It repeats; a bar shows the repeated digit: $-0.\bar{6}$. Every rational number has a decimal form that ends or eventually repeats.

Do not round a repeating decimal too early when comparing close values. A common denominator can give an exact comparison without rounding.

Convert a decimal to a fraction

Place value tells the denominator. For negative 1.25, there are 125 hundredths:

Simplify by dividing the numerator and denominator by the same common factor. The sign stays attached. For example, dividing both parts of $-\frac{18}{24}$ by 6 gives $-\frac{3}{4}$.

Compare matching forms

Of two negative numbers, the one with the larger absolute value is less. Compare $-\frac{3}{4}$ and $-\frac{2}{3}$ using twelfths:

Nine twelfths below zero lies left of eight twelfths below zero. Decimals tell the same story: $-0.75<-0.666\dots$.

For $-0.5$ and $-0.45$, write $-0.50$. Fifty hundredths below zero is farther left than forty-five hundredths below zero, so $-0.5<-0.45$. Comparing “5” with “45” ignores place value.

Opposites and absolute value still work: $-(-\frac{2}{5})=\frac{2}{5}$ and $\mathrm{\mid }-\frac{5}{6}\mathrm{\mid }=\frac{5}{6}$. The fraction changes how distance is measured, not what an opposite or distance means.

Fraction parts must have the same size before you combine them. Decimal places must line up. After that, use the sign rules for addition and subtraction.

Add fractions with different signs

To find $-\frac{2}{3}+\frac{1}{4}$, use twelfths. Thirds and fourths can both be split into twelfths:

Now combine the signed numerators and keep the common denominator:

The negative part is larger, so the answer is negative. Adding the denominators would change the size of the parts. For example, $-\frac{1}{2}+\frac{1}{3}$ is $-\frac{1}{6}$, found using sixths; adding the tops and bottoms would wrongly give zero.

Subtract a fraction

Subtraction still means adding the opposite. A subtraction can cross zero:

When the second fraction is negative, change it to its positive opposite first:

Using tenths gives $-\frac{6}{10}+\frac{5}{10}=-\frac{1}{10}$. Removing a negative amount makes the starting value greater, even though the final answer remains negative.

Keep mixed-number signs together

Convert mixed numbers to improper fractions when the whole-number and fraction parts are awkward to handle separately:

Both amounts contribute negatively to the result. A decimal check is $-1.5-2.75=-4.25$.

Line up decimal places

For $-4.7+2.35$, write $-4.70+2.35$. The negative size is larger. Subtract $2.35$ from $4.70$ to get $2.35$, then keep the negative sign. The sum is $-2.35$.

For subtracting a negative decimal:

The opposite of $-1.25$ is $1.25$. Keeping the first number negative matters; changing both signs would produce a different answer.

Estimate before using exact fractions

For $-3\frac{7}{8}+1\frac{1}{10}$, nearby integers give $-4+1=-3$. The exact calculation is

That is close to the estimate and negative. In a money example, changes of −$1.35 and +$0.80 total −$0.55: a fall of 55 cents.

Decide the sign first. Then find the size using positive fraction or decimal methods. The sign rules do not change when a value lies between integers.

Multiply fractions

Multiply the numerators and multiply the denominators, then simplify:

You can simplify factors before multiplying. Divide the 9 and 3 by 3, and the 2 and 10 by 2. This leaves a numerator size of 3 and denominator size of 5. Different signs make the result negative.

Convert mixed numbers first:

The whole-number and fraction parts are one value. Multiplying those parts separately loses products needed for the full amount.

Divide using a reciprocal

The reciprocal of a nonzero number multiplies with it to give 1. Swap the numerator and denominator, keeping the sign:

Both factors are negative, so their product is positive 1. Using positive three halves would give negative 1 and would not be the reciprocal.

To divide by a fraction, multiply by the reciprocal of the divisor:

Multiplying by a fraction and then by its reciprocal gives a factor of 1, undoing the first multiplication. A quotient must multiply by the original divisor to return the dividend, so multiplying the dividend by the reciprocal finds that missing factor.

Only the divisor is replaced by its reciprocal. The dividend stays as it was. Check by multiplying the result by the original divisor: $\frac{3}{2}\times (-\frac{1}{2})=-\frac{3}{4}$.

A complex fraction has a fraction above or below its main fraction bar. Read that main bar as division:

Place a decimal point

For $(-1.2)(-0.5)$, the signs give positive. Multiply sizes to get $0.6$. Squaring negative three tenths gives $(-0.3{)}^2=0.09$, since $0.3\times 0.3=0.09$.

For $-2.4÷(-0.06)$, multiply both numbers by 100 to remove the divisor’s decimal:

Scaling both numbers equally keeps the quotient unchanged. Scaling only the divisor would change the answer.

Check what multiplication does to a negative value

Multiplying a nonzero number by a positive fraction below 1 reduces its absolute value. For a negative number, that moves it closer to zero and makes it greater:

Use a size estimate for less friendly values. For $(-48.7)\times 3.1$, estimate $(-50)\times 3=-150$. The exact result, $-150.97$, has the expected sign and size.

Negative numbers follow the usual order of operations. Parentheses can change the order, so read the grouping before starting.

Use equal priority correctly

Work in this order:

1. Calculate inside grouping symbols, starting with the innermost group.
2. Evaluate exponents.
3. Multiply and divide from left to right.
4. Add and subtract from left to right.

PEMDAS names these stages. GEMDAS, BODMAS, BIDMAS, and BEDMAS name the same order. Multiplication does not outrank division; addition does not outrank subtraction.

For $8-2\times (-3)$, multiply first:

Doing the subtraction first would answer $(8-2)\times (-3)$, a different expression.

With division and multiplication, follow their written order:

Multiplying $4$ and $-2$ first would insert grouping that is not present.

Keep the power’s base clear

In $-3^2$, square 3 before taking its opposite, giving $-9$. In $(-3{)}^2$, the whole negative value is squared, giving $9$.

For $5-(-2{)}^3$, the cube is $-8$. Then $5-(-8)=13$. Work out the power before handling the subtraction sign beside it.

Treat a fraction bar as grouping

Everything above the bar forms the numerator. Everything below forms the denominator. Calculate each group before dividing:

The subtraction belongs in the numerator, and the multiplication belongs in the denominator. Dividing only one term would lose that grouping.

For nested parentheses, work outward one layer at a time:

The inner subtraction gives $-3$. Squaring gives $9$. The outer parentheses give $5-9=-4$. Finally, $-2\times (-4)=8$.

Distribute or calculate inside first

For $-3(4-7)$, calculating inside gives $-3(-3)=9$. Distribution gives the same result:

The positive 21 comes from multiplying negative three by negative seven. The outside factor must multiply both parts.

Enter negative values on a calculator

Use the negative-number key, often marked (−) or +/−, to enter a negative value. Use subtraction to subtract one value from another. To square the whole negative number, enter parentheses around it before the exponent. Compare the displayed expression with the written one, then check the expected sign. For $(-3{)}^2$, the answer must be positive.

Substitution means replacing a variable with its value. Put a negative replacement in parentheses so its sign stays attached to the whole value.

Replace every occurrence

If $x=-2$:

The first expression multiplies by 5. The second subtracts 7. Substitution keeps those operations in place; it only replaces the letter.

Evaluate the power before a multiplying number

For $3x^2$ with $x=-2$, substitute first:

Only the value of $x$ is squared. Squaring $3(-2)$ would instead give $36$. Leaving out the parentheses and writing $-2^2$ would give the wrong sign for the square of $x$.

An outside minus sign still acts afterward. If $x=-5$, then $-x=5$, but $-x^2=-25$. The square is 25; its opposite is negative 25.

Keep two variables in their places

Let $a=-3$ and $b=4$. Then $a-b=-3-4=-7$, while $b-a=4-(-3)=7$. Their product is $ab=(-3)(4)=-12$.

In $a^{2}b$, square the replacement for $a$ first, then multiply by $b$: $9\times 4=36$. For a fraction expression:

Keep the whole numerator together. The fraction bar groups it before division.

Substitute into a temperature formula

The Celsius-to-Fahrenheit formula is $F=1.8C+32$. With $C=-15$:

So $-15$ °C equals $5$ °F. A negative input need not give a negative output because the formula also adds 32. At $C=-40$, the formula gives $-72+32=-40$, where the scales agree.

The reverse formula is $C=\frac{5}{9}(F-32)$. For $F=-4$, first calculate the parentheses:

Thus $-4$ °F equals $-20$ °C.

Fractions, decimals, and several terms

If $x=-\frac{1}{2}$, then $4x+1=4(-\frac{1}{2})+1=-1$. Its square is $x^2=\frac{1}{4}$. For $y=-0.4$, calculate $10y^2=10(0.16)=1.6$.

For $x=-1$ in $x^2-5x+6$, calculate each term with its sign:

A square cannot be negative. That sign check helps catch missing parentheses before a wrong square affects the rest of the expression.

A multi-step problem asks you to connect the numbers before calculating. Decide what zero means and which direction is positive. Then write an expression that includes every change.

Find a mean with the signs included

The mean is the sum divided by the number of values. Five daily low temperatures are $-3$, $2$, $-5$, $-6$, and $2$ °C.

The positive readings total 4 and the negative readings total negative 14. The sum is $-10$, so

The mean is $-2$ °C. Adding absolute values would measure distance from zero and lose which readings were below zero.

Combine a starting value with a rate

A freezer starts at $5$ °C and cools steadily by 2.5 degrees per hour. Its signed rate is $-2.5$ degrees per hour. After 6 hours:

The temperature is $-10$ °C. The product $-15$ is the change, so you still need to add the starting value.

To find the time needed to reach $-18$ °C, first find the required change:

Then divide by the rate: $-23÷(-2.5)=9.2$ hours. Check by calculating $5+9.2(-2.5)=-18$.

Keep income and costs separate

A club starts with a balance of −$120. It sells 45 tickets at $4 each and pays $35 for supplies. Write

Ticket income is $45\times 4=180$ dollars. The new balance is $-120+180-35=25$ dollars. The club has $25 after paying what it owed and buying the supplies.

Follow changes in depth

A diver starts at $-6$ meters and descends 1.5 meters per minute for 8 minutes. The descent is $8(-1.5)=-12$ meters, putting the diver at $-18$ meters. Rising 7.5 meters then gives $-18+7.5=-10.5$ meters: 10.5 meters below the surface.

Apply scoring rules

A quiz awards 3 points per right answer, loses 2 per wrong answer, and gives zero for a blank. With 12 right, 5 wrong, and 3 blank:

The score is 26 points. The count of wrong answers is positive; the negative score per wrong answer makes their total contribution negative.

Check the result

Check the sign against the story. Compare the size with a rough estimate. Work backward when possible, as in the freezer calculation. A final sentence should identify the requested quantity: “9.2 hours,” “the club has $25,” or “10.5 meters below the surface.”