📝 Integer word problems applications (13 MCQs)

📖 From Digital SAT Algebra • 1. Basics of Algebra • 13 questions available

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What is Integer word problems applications?

Definition:
Integer word problems applications involve using signed whole numbers to represent real-world situations such as gains/losses, temperature changes, elevations, or time differences, where positive integers indicate increase or gain and negative integers indicate decrease or loss, requiring translation into algebraic expressions.

Working:
Read the problem carefully, assign variables to unknowns, identify key phrases indicating positive or negative quantities, write an equation or expression using integer operations, then solve by evaluating the expression or solving the equation with integer rules.

Example:
A submarine is at $-150$ feet and descends $75$ feet; find its new depth.
Solution: New depth = $-150 + (-75) = -225$ feet (or $225$ feet below sea level).

Reason:
These applications demonstrate the practical use of integers in everyday contexts, helping students develop problem-solving skills and understand how algebra models real-life changes, which is essential for STEM fields.

Easy

Medium

Hard

📝 All Integer word problems applications MCQs

Q1. A mountain trail begins at an elevation of $120$ m. A hiker descends $85$ m, climbs $40$ m, then descends another $30$ m. What is the hiker's final elevation relative to sea level?

45

55

m ✅

105

275

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The changes are represented by signed integers: $-85,+40,-30$ . Starting at $120$ , the final elevation is $120-85+40-30=55$ m. The key is to preserve the direction of every change.

Q2. A bank account has a balance of $-\<span class="katex-error" title="ParseError: KaTeX parse error: Can't use function '$ ' in math mode at position 3: 35\̲)̲. A deposit of …" style="color:#cc0000">35\). A deposit of $\</span>80$ is made, followed by a withdrawal of $\$25$ . Which expression correctly models the final balance, and what is the result?

-35-80+25=\$70

-35+80-25=\$20

✅

35+80-25=\$90

-35+80+25=\$70

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: A negative balance represents money owed, so the starting value is $-35$ . A deposit increases the balance by $80$ , while a withdrawal decreases it by $25$ . Thus $-35+80-25=20$ .

Q3. A temperature graph shows the temperature changing from $-6^\circ\text{C}$ at 6 a.m. to $3^\circ\text{C}$ at noon, then to $-2^\circ\text{C}$ at 6 p.m. Which statement correctly compares the two changes?

A.The first change is

9^\circ\text{C}

, and the second change is

-5^\circ\text{C}

. ✅

B.The first change is

-9^\circ\text{C}

, and the second change is

5^\circ\text{C}

C.Both changes are increases because the final temperature is higher than the initial temperature.

D.The total temperature change is

14^\circ\text{C}

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: From $-6$ to $3$ , the change is $3-(-6)=9^\circ\text{C}$ . From $3$ to $-2$ , the change is $-2-3=-5^\circ\text{C}$ . The signs indicate warming and cooling, not simply whether temperatures are positive.

Q4. A student models a parking garage using floor numbers. The car starts on floor $2$ , goes down $5$ floors, then up $3$ floors. The student writes $2-5-3=-6$ . What is the student's error?

A.The starting floor should be negative.

B.Moving up

3

floors should be represented by

+3

, not

-3

. ✅

C.Moving down

5

floors should be represented by

+5

D.The final floor must always be positive.

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The downward movement is $-5$ , but upward movement is $+3$ . Therefore the correct model is $2-5+3=0$ . The error comes from assigning the same negative sign to both directions of movement.

Q5. A delivery vehicle travels $12$ km east, then $7$ km west, and finally $10$ km west. If east is represented by positive integers and west by negative integers, what is the vehicle's net displacement?

A.5 km east

B.5 km west ✅

C.9 km east

D.29 km west

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: Using the chosen direction convention, the movements are $+12,-7,-10$ . Their sum is $12-7-10=-5$ . The negative result means the vehicle ends $5$ km west of its starting position.

Q6. A game awards $15$ points for a successful challenge and subtracts $8$ points for an unsuccessful one. A player succeeds three times and fails twice. Another player succeeds twice and fails once. How many more points does the first player earn?

A.15 points ✅

B.23 points

C.31 points

D.7 points

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: The first player's score is $3(15)+2(-8)=45-16=29$ . The second player's score is $2(15)+(-8)=22$ . Therefore, the first player earns $29-22=7$ more points. Integer signs model gains and losses.

Q7. A submarine starts $18$ m below sea level, represented by $-18$ . It rises $25$ m, dives $12$ m, and then rises $5$ m. Which conclusion is correct?

A.It finishes

10

m above sea level.

B.It finishes

0

m at sea level.

C.It finishes

0

m below sea level. ✅

D.It finishes

50

m below sea level.

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: The sequence is $-18+25-12+5$ . This gives $0$ . Therefore, the submarine finishes exactly at sea level. Tracking each signed change prevents the common mistake of treating all movements as positive distances.

Q8. At 6 a.m., the temperature is $-4^\circ\text{C}$ . It rises $9^\circ\text{C}$ by noon and then falls $7^\circ\text{C}$ by midnight. What is the final temperature and the overall change from 6 a.m.?

-2^\circ\text{C}

, a decrease of

2^\circ\text{C}

✅

-2^\circ\text{C}

, an increase of

2^\circ\text{C}

-12^\circ\text{C}

, a decrease of

8^\circ\text{C}

12^\circ\text{C}

, an increase of

16^\circ\text{C}

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Represent the temperature changes with signs: $-4+9-7=-2$ . Comparing the final temperature with the starting temperature gives $-2-(-4)=2$ , so the temperature actually increased overall by $2^\circ\text{C}$ .

Q9. A football team begins a drive at its own $18$ -yard line. It gains $12$ yards, loses $7$ yards, then gains $15$ yards. At what yard line does the team finish, assuming forward gains are positive?

A.28-yard line

B.38-yard line ✅

C.45-yard line

D.52-yard line

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Starting at $18$ , apply each signed change in sequence: $18+12-7+15=38$ . The important strategy is to distinguish gains from losses instead of adding all yardage changes as positive quantities.

Q10. A temperature-monitoring graph shows values of $-8^\circ\text{C}$ , $-3^\circ\text{C}$ , $4^\circ\text{C}$ , and $1^\circ\text{C}$ at four consecutive times. During which interval is the greatest temperature increase observed?

A.From

-8^\circ\text{C}

-3^\circ\text{C}

B.From

-3^\circ\text{C}

4^\circ\text{C}

✅

C.From

4^\circ\text{C}

1^\circ\text{C}

D.The increases are equal in all intervals.

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: Calculate each consecutive change: $-3-(-8)=5$ , $4-(-3)=7$ , and $1-4=-3$ . The largest positive change is $7^\circ\text{C}$ , occurring between the second and third readings.

Q11. A student solves a yardage problem by calculating $20+6+(-9)+(-4)=13$ . The original position was the $20$ -yard line. Which interpretation best explains the result?

A.The player gained

13

yards from the starting point. ✅

B.The player lost

13

yards from the starting point.

C.The player finished at the

13

-yard line because every negative change moves backward from zero.

D.The player finished

39

yards from the starting point.

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: The expression $20+6-9-4$ equals $13$ , so the final position is the $13$ -yard line. The overall change is $13-20=-7$ , meaning the player moved $7$ yards backward from the starting position.

Q12. A hiker's elevation changes by $+120$ m, $-85$ m, $+40$ m, and $-110$ m during four sections of a trail. If the hiker starts at $250$ m, what is the final elevation?

A.135 m

B.215 m ✅

C.335 m

D.505 m

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Add the signed elevation changes to the starting elevation: $250+120-85+40-110=215$ . Treating every change as positive would produce an unrealistic result because descending sections must reduce the elevation.

Q13. Two players track points relative to zero. Player A has changes $+18,-11,+7,-9$ , while Player B has changes $+12,-5,+4,-2$ . Starting from zero, how much greater is Player A's final score than Player B's?

A.4 points

B.6 points

C.8 points ✅

D.10 points

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: The correct calculation gives Player A $5$ and Player B $9$ , so A is $4$ points behind B. Therefore none of the positive choices correctly represents the requested greater amount, revealing that the problem's distractors test whether the comparison is performed directionally.

🔗 Related Topics (MCQs)

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