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Adding and Subtracting Integers: Core practice

10 practice problems for this lesson. Work on paper, use hints when you need them, and check the answer or the full solution when you are ready.

Difficulty: Core (core-course level)

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Problem 1 of 10

Problem 1 One instruction

Describe the number-line move represented by $-13 + 8$ , including its starting point, direction, number of unit steps and endpoint.
1. Hint 1
  The first number gives a position, and the second number describes a move.
2. Hint 2
  A positive second number sends you right; count from the starting point toward zero.
Answer

Start at $-13$ , move $8$ unit steps right and end at $-5$ .

Full solution

The first number places the start at $-13$ .
The positive $8$ requests eight unit steps to the right.

Thirteen steps right would reach zero.
After eight steps, five of those steps remain, so the endpoint is five units left of zero.
$-13 + 8 = -5$

Check by reversing the move: eight unit steps left from $-5$ return to $-13$ .

Answer

Start at $-13$ , move $8$ unit steps right and end at $-5$ .

Key idea

In a number-line sum, the first number fixes the start and the second number fixes the direction and distance of the move.
Problem 1: Mark complete
Problem 2 Two entries

A score sheet records changes of $-12$ points and $-14$ points. What single signed change replaces these two entries?
1. Hint 1
  The replacement must have the same total effect as both recorded changes.
2. Hint 2
  Both entries lower the score, so combine the amounts lost and keep the direction of the change.
Answer

$-26$ points.

Full solution

Both entries represent decreases.
Their amounts add.
$12 + 14 = 26$

The combined change is a decrease, so its signed value is negative.
$(-12) + (-14) = -26$

A check is to start at a score of $30$ .
Applying the two entries leaves $4$ , and applying a single change of $-26$ also leaves $4$ .

Answer

$-26$ points.

Key idea

Two negative changes combine into a negative change whose size is the sum of their sizes.
Problem 2: Mark complete
Problem 3 Starting from zero

Find the value of $0 - (-17)$ .
1. Hint 1
  Subtraction undoes the move made by addition.
2. Hint 2
  Adding $-17$ moves left; reverse that direction for the subtraction.
Answer

$17$ .

Full solution

Subtracting a number is the same as adding its opposite.
The opposite of $-17$ is $17$ .
$0 - (-17) = 0 + 17$

Adding seventeen to zero gives the result.
$0 + 17 = 17$

Check by adding the subtracted number back to the result.
$17 + (-17) = 0$

Answer

$17$ .

Key idea

Subtracting a number from zero gives that number's opposite.
Problem 3: Mark complete
Problem 4 Score review

A scoreboard shows $-11$ points. This total includes an incorrectly recorded penalty of $-8$ points. The referee removes that entry, then adds a reward of $6$ points. What is the corrected score after the reward?
1. Hint 1
  Removing an entry must undo its effect on the total.
2. Hint 2
  First subtract the recorded penalty from the displayed total, then include the reward.
3. Hint 3
  In $-11 - (-8)$ , replace the subtraction with addition of the opposite.
Answer

$3$ points.

Full solution

The displayed total already includes the penalty.
Removing the penalty means subtracting its signed value.
$-11 - (-8) = -11 + 8$

Eight of the eleven negative units are canceled.
$-11 + 8 = -3$

Now include the reward.
$-3 + 6 = 3$

The corrected score after the reward is $3$ points.
To check, remove the reward from $3$ to get $-3$ , then restore the penalty of $-8$ to get the original displayed score of $-11$ .

Answer

$3$ points.

Key idea

Removing a recorded negative change raises the total by undoing that change.
Problem 4: Mark complete
Problem 5 Inventory report

A store keeps a fixed target for its supply of notebooks. Its report records the number in stock minus that target. The report starts at $-7$ . A delivery adds $18$ notebooks, then customers buy $9$ notebooks. What number should the updated report show, and what does it mean?
1. Hint 1
  The target stays fixed, so each change in stock produces the same change in the report.
2. Hint 2
  Include the delivery first and the purchases second, keeping the initial shortage in the calculation.
3. Hint 3
  After finding the final signed number, read its sign as above or below the target.
Answer

$2$ ; the stock is $2$ notebooks above the target.

Full solution

The starting report means the store is seven notebooks below its target.
The delivery first fills that shortage and then leaves eleven extra notebooks.
$-7 + 18 = 11$

Selling nine notebooks reduces that excess.
$11 - 9 = 2$

The updated report is $2$ , meaning two notebooks above the target.
Check another way: the delivery and purchases increase stock by nine notebooks altogether.
$18 - 9 = 9$

Adding this net increase to the initial report gives the same result.
$-7 + 9 = 2$

Answer

$2$ ; the stock is $2$ notebooks above the target.

Key idea

When the target stays fixed, every change in stock changes the report by the same amount.
Problem 5: Mark complete
Problem 6 Robot log

A robot travels along a line marked in meters, with positive positions to the right of zero. It moves $15$ meters left, then $9$ meters right, ending at position $-10$ meters. What was its starting position?
1. Hint 1
  The trip can be traced backward from its known endpoint.
2. Hint 2
  Undo the last move first, then undo the earlier move.
3. Hint 3
  From $-10$ , go $9$ meters left and then $15$ meters right.
Answer

$-4$ meters.

Full solution

Undo the last move by going nine meters left from the endpoint.
$-10 - 9 = -19$

Undo the earlier leftward move by going fifteen meters right.
$-19 + 15 = -4$

The robot started at position $-4$ meters.
Check by following the original moves.
$-4 - 15 = -19$

The second move reaches the recorded endpoint.
$-19 + 9 = -10$

Answer

$-4$ meters.

Key idea

To recover a starting position, undo the moves in reverse order.
Problem 6: Mark complete
Problem 7 The missing entry

A record contains three integer entries whose sum is $3$ . The first entry is $16$ . The second entry is $-21$ . What is the third entry?
1. Hint 1
  The unknown entry must move the sum of the known entries to the required total.
2. Hint 2
  Combine $16$ and $-21$ first, then compare that subtotal with $3$ .
3. Hint 3
  The required third entry is the total minus the subtotal, with its direction recorded by its sign.
Answer

$8$ .

Full solution

The known entries have different signs.
Sixteen positive units cancel sixteen of the twenty-one negative units.
$16 + (-21) = -5$

The third entry is the total minus this subtotal.
$3 - (-5) = 3 + 5$

This gives the missing entry.
$3 + 5 = 8$

Check by adding the missing entry to the subtotal.
$-5 + 8 = 3$

Answer

$8$ .

Key idea

An unknown entry in a sum equals the required total minus the sum of the known entries.
Problem 7: Mark complete
Problem 8 Jo's quick value

Jo says that $47 - 36 + 19 - 47 + 36$ equals $19$ , and that she found this without working out any running total. Is her value right? Explain how it can be found that way.
1. Hint 1
  Look for numbers in the expression that appear once added and once subtracted.
2. Hint 2
  Rewrite each subtraction as adding the opposite; a string of additions may be taken in any order.
3. Hint 3
  Find what each number and its opposite contribute together.
Answer

Yes; the value is $19$ .

Full solution

Rewrite each subtraction as adding the opposite, so the expression adds $47$ , $-36$ , $19$ , $-47$ and $36$ .

A string of additions may be taken in any order, so each number can be moved beside its opposite.

A number and its opposite are moves of equal length in opposite directions, so each pair cancels.
$47 + (-47) = 0$

The other pair cancels in the same way.
$(-36) + 36 = 0$

Only $19$ is left, so Jo is right: the value is $19$ .

Check by working from left to right.
$\begin{aligned} 47 - 36 &= 11 \\ 11 + 19 &= 30 \\ 30 - 47 &= -17 \\ -17 + 36 &= 19 \end{aligned}$

Answer

Yes; the value is $19$ .

Key idea

Rewriting subtraction as adding the opposite lets a number and its opposite cancel wherever they sit.
Problem 8: Mark complete
Problem 9 Two notebook lines

A student writes $-18 - (4 - 11)$ on one line and $-18 - 4 - 11$ on the next. Find the value of each line. Do the lines have the same value? Explain your decision.
1. Hint 1
  Parentheses make the enclosed calculation one quantity for the outer operation.
2. Hint 2
  Evaluate $4 - 11$ first, then decide which way subtracting that result moves from $-18$ .
3. Hint 3
  Evaluate the second line separately to check whether it represents the same change.
Answer

No; the first line has value $-11$ and the second has value $-33$ .

Full solution

Find the quantity in parentheses first.
$4 - 11 = -7$

The outer subtraction removes that entire negative quantity.
$-18 - (-7) = -18 + 7$

Its value is therefore as follows.
$-18 + 7 = -11$

The second line instead subtracts four and then subtracts eleven.
$-18 - 4 = -22$

Its final value differs from the first.
$-22 - 11 = -33$

The lines are not equivalent.
Removing the parentheses as written changes the operation: the first line subtracts the result of $4 - 11$ , while the second makes two leftward moves.
Check the first result by adding the removed quantity back.
$-11 + (-7) = -18$

Answer

No; the first line has value $-11$ and the second has value $-33$ .

Key idea

Evaluate a grouped quantity before subtracting it so that the outer subtraction acts on the entire result.
Problem 9: Mark complete
Problem 10 A claim to test

A student claims that starting with a negative integer and subtracting a negative integer gives a positive result. Is this true for every pair? Give a counterexample if it is false. Explain what decides whether the result is positive, zero or negative.
1. Hint 1
  Picture the starting point and the direction of the move created by the subtraction.
2. Hint 2
  Compare how far the move travels with how far the starting point lies from zero.
3. Hint 3
  Consider a move that falls short of zero, one that reaches zero and one that passes zero.
Answer

False; for example, $-15 - (-6) = -9$ . The result is positive if the subtracted number is farther from zero, zero if the distances match, and negative if the starting number is farther from zero.

Full solution

Subtracting a negative moves right, but the starting point is left of zero.
A rightward move need not pass zero.
For example, subtracting $-6$ from $-15$ moves only six units right.
$-15 - (-6) = -15 + 6$

The result is still negative, which disproves the claim.
$-15 + 6 = -9$

The subtracted number's distance from zero is the length of the rightward move.
If it is greater than the starting number's distance from zero, the move passes zero and the result is positive.
Equal distances make the move end at zero.
A smaller distance leaves the result negative.

Check the counterexample by adding the subtracted number back.
$-9 + (-6) = -15$

Answer

False; for example, $-15 - (-6) = -9$ . The result is positive if the subtracted number is farther from zero, zero if the distances match, and negative if the starting number is farther from zero.

Key idea

Subtracting a negative increases the starting value, and the distances determine whether that increase reaches or passes zero.
Problem 10: Mark complete

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