7th Grade · 7.NS.A.3

Integer word problems in 7th grade

Integer word problems are real situations that use positive and negative numbers, such as a depth below sea level or a loss of points. In 7th grade, your child solves them with all four operations: adding, subtracting, multiplying and dividing. The numbers can be integers, fractions or decimals.

What the standard says

The official Common Core wording, quoted exactly. The plain version follows.

7.NS.A.3 Solve real-world and mathematical problems involving the four operations with rational numbers.

What it means, in plain words

A rational number is any number that can be written as a fraction of two integers, such as 5, $-3$ , $\frac{2}{3}$ or $-1.5$ . In 7th grade, operations with rational numbers follow the same sign rules as integers. Adding a negative number lowers a total, and subtracting a negative number raises it. Multiplying or dividing two negative numbers gives a positive answer.

The hard part of integer word problems is the words. Your child decides which way each number points. Below, loss, owe, drop and deeper usually mean negative. Above, gain, earn and rise usually mean positive. Then your child writes one expression, works it out and says the answer in words, with its unit.

Rational numbers word problems also bring in complex fractions. A complex fraction has a fraction on top, on the bottom or both, and it is a division: $\frac{\frac{3}{5}}{\frac{1}{10}}$ means $\frac{3}{5} \div \frac{1}{10}$ , which is 6. A quick estimate checks the answer. If the story describes a small loss, an answer that shows a big gain needs a second look.

Your child should be able to:

Turn words such as below, loss and deeper into negative numbers
Write one expression for a problem with several steps
Add, subtract, multiply and divide positive and negative fractions and decimals
Check that the answer makes sense and state it in words with its unit

Example

A diver is 18 feet below the surface. The diver rises 7 feet, then goes 12 feet deeper. How far below the surface is the diver now?

Below the surface is negative, so the start is $-18$ . Rising is $+7$ , and going deeper is $-12$ .
$-18 + 7 - 12 = -11 - 12 = -23$
The diver is 23 feet below the surface.

Example

The water in a pond drops 1.5 inches each week for 4 weeks. Then a storm raises it 2.25 inches. What is the total change in the water level?

Each weekly drop is $-1.5$ . Four drops: $4 \times (-1.5) = -6$ .
The storm adds 2.25: $-6 + 2.25 = -3.75$ .
The water level ends 3.75 inches lower than at the start.

Where children usually get stuck

For the diver, your child adds every number and gets 37 feet.
The signs are in the words (below, rises, deeper), and the child reads only the numbers.
What helps. Before any arithmetic, ask your child to write a plus or a minus sign next to each number and say which word gave it.
A submarine goes from 40 feet below the surface to 15 feet below. Your child works out $-40 - (-15) = -25$ and says it rose $-25$ feet.
A change is the end minus the start. The child subtracted in the wrong order. The right calculation is $-15 - (-40) = 25$ , so the submarine rose 25 feet.
What helps. Sketch a vertical number line with 0 at the surface. Mark both depths, count the steps between them and ask whether the submarine went up or down.
Your child writes that the diver is " $-23$ feet below the surface".
The minus sign and the word "below" carry the same direction. Using both turns the answer back into 23 feet above the surface.
What helps. Ask your child to say the answer as a sentence a person outside class would understand: "The diver is 23 feet below the surface."

Try four practice questions

Real 7th grade questions from our question bank, easiest first. Answer them right here, check each one, and see a worked solution for any question.

1Warm-up
The vertical number line shows the starting temperature. It rises by $\frac{1}{2}$ ° each hour for 2 hours, then drops $\frac{1}{2}$ °. Write the final temperature as a fraction, mixed number, or decimal.
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Show the answer and how to solve it
Answer. 1
The final temperature is $1$ °.
1. 1Look at the vertical number line. The starting temperature is $\frac{1}{2}$ °.
2. 2The temperature rises $\frac{1}{2}$ ° each hour for 2 hours. Find the total rise first.
3. 3Add the total rise to the starting temperature, then subtract the final drop.
Common mistake. A common mistake is counting the $\frac{1}{2}$ ° rise only once instead of for both hours. The total rise is $2 \times \frac{1}{2} = 1$ °. Add it to the starting temperature of $\frac{1}{2}$ °, then subtract the $\frac{1}{2}$ ° drop.
The idea underneath. A repeated signed change can be combined into one total change first, then added to the starting value on the number line.
2On level
A hiker walks $\frac{5}{8}$ mile in $\frac{1}{2}$ hour. At the same rate, how far would the hiker walk in 1 hour? Select all choices that show the distance.
A
$\displaystyle \frac{5}{16}$
B
$\displaystyle 1.25$
C
$\displaystyle \frac{4}{5}$
D
$\displaystyle 0.8$
E
$\displaystyle \frac{5}{4}$
Show the answer and how to solve it
The correct choices are $\frac{5}{4}$ and 1.25 because $\frac{5}{8} \div \frac{1}{2} = \frac{5}{4}$ , and $\frac{5}{4}$ is the same as 1.25.
1. 1The hiker walks $\frac{5}{8}$ mile in $\frac{1}{2}$ hour. One hour is $2$ times as long. At the same rate, multiply the distance by $2$ too. This is the same as dividing by $\frac{1}{2}$ .
2. 2To divide by a fraction, multiply by its reciprocal.
3. 3Convert the reduced fraction to a decimal.
4. 4So the two correct choices are $\frac{5}{4}$ and 1.25.
Common mistake. A common mistake is reversing the division and finding $\frac{1}{2} \div \frac{5}{8} = \frac{4}{5}$ . Another is multiplying to get $\frac{5}{16}$ , but dividing by $\frac{1}{2}$ should make the result greater than $\frac{5}{8}$ , not smaller.
The idea underneath. Dividing by a positive number less than 1 makes the quotient larger, because you are counting how many smaller pieces fit into the amount.
3On level
The temperature changed by $-\frac{5}{4}$ ° in $\frac{1}{2}$ hour. At the same rate, what was the change per hour? Write the answer as a fraction or decimal.
Loading...
Show the answer and how to solve it
Answer. -\frac{5}{2}
The answer is $-\frac{5}{2}$ , or $-2.5$ , because a $-\frac{5}{4}$ ° change in half an hour becomes a $-\frac{5}{2}$ ° change in one full hour.
1. 1The temperature changed by $-\frac{5}{4}$ ° in $\frac{1}{2}$ hour. To find the change in 1 full hour, divide the total change by the time.
2. 2Dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of $\frac{1}{2}$ is $\frac{2}{1}$ .
3. 3Multiply and simplify. The answer stays negative because a negative number divided by a positive number is negative. Both forms are accepted: $-\frac{5}{2}$ or $-2.5$ .
Common mistake. A common mistake is reversing the quantities and computing $\frac{1}{2} \div \left(-\frac{5}{4}\right)$ instead. Always divide the total change by the time, not the other way around.
The idea underneath. A per-unit question means total amount divided by number of units, and dividing by $\frac{1}{2}$ doubles the amount because two halves make one whole.

4Stretch

Look at Marcus's work below. Which mistake best explains the error?

	$\displaystyle -6 \div \frac{3}{2}$
Step 1	$\displaystyle = -6 \times \frac{3}{2}$
Step 2	$\displaystyle = -\frac{18}{2}$
Step 3	$\displaystyle = -9$

The student divided by the fraction by multiplying by it instead of its reciprocal.

The student added or subtracted before dividing.

The student treated subtracting a negative like subtracting a positive.

The student simplified

-\frac{18}{2}

incorrectly.

Show the answer and how to solve it

The correct choice is 'The student divided by the fraction by multiplying by it instead of its reciprocal' because $-6 \div \frac{3}{2}$ should equal $-6 \times \frac{2}{3} = -4$ , not $-9$ .

1Inspect Marcus's work line by line.
2The original expression $-6 \div \frac{3}{2}$ is written correctly, so the starting point is fine.
3The error is in Step 1. Dividing by a fraction requires multiplying by its reciprocal. The reciprocal of $\frac{3}{2}$ is $\frac{2}{3}$ , not $\frac{3}{2}$ itself.
4Now multiply using the correct reciprocal.
5The correct result is $-4$ , not $-9$ . Marcus multiplied by $\frac{3}{2}$ instead of flipping it to $\frac{2}{3}$ first.

Common mistake. A common mistake when dividing by a fraction is to multiply by the same fraction rather than flipping it. To divide by $\frac{3}{2}$ , students must use the reciprocal $\frac{2}{3}$ .

The idea underneath. To divide by a fraction, multiply by its reciprocal. Check the first line where division changes to multiplication.