Word problems with integers

Integers shine when situations have natural "negatives" , temperatures below zero, debts, losses, depths below sea level.

Idea

Common contexts and the integer they suggest:

• Temperature. $-5\,^\circ$C means $5$ degrees below freezing.
• Money. $-\rupee 200$ means a debt of $\rupee 200$.
• Elevation. $-50$ m means $50$ m below sea level.
• Sports. $-3$ in a handicap or net score below par.
• Time. $-2$ hours could mean $2$ hours ago.

Once you assign signs, operations become natural. A temperature rise is positive; a fall is negative. A profit is positive; a loss is negative.

Multiplication often appears as a "rate × time" calculation. If a glacier melts $2$ m per year (a $-2$ m change in height per year) for $5$ years, the total change is $-2 \times 5 = -10$ m.

Division appears as a "per unit" calculation. If the temperature dropped by $36^\circ$ in $4$ hours, the average drop per hour is $-36 \div 4 = -9$, i.e., $9^\circ$ per hour drop.

Worked examples

Example 1. Temperature was $-3\,^\circ$C at $6$ am. It dropped $2^\circ$ each hour for $4$ hours. Temperature at $10$ am?

Net change: $4 \times (-2) = -8$. Final: $-3 + (-8) = -11\,^\circ$C.

Example 2. A submarine descends $5$ m per minute. Starting from sea level ($0$), where is it after $9$ minutes?

Descent of $5$ m per minute means change $-5$ per minute. After $9$ minutes: $9 \times (-5) = -45$ m. So $45$ m below sea level.

Example 3. A shopkeeper made losses of $\rupee 50$ per day for $7$ days. Total loss?

Loss: $7 \times (-50) = -\rupee 350$.

Example 4. A factory's profit dropped by $\rupee 1200$ over $8$ months. Average monthly change in profit?

$-1200 \div 8 = -150$. So profit dropped by $\rupee 150$ per month on average.

Try it yourself

1. Temperature rose $4^\circ$ per hour for $3$ hours, starting at $-7\,^\circ$C. New temperature?
2. A diver descends $4$ m per second for $6$ s. Where is she now (relative to surface)?
3. A bank account starts at $-\rupee 500$. Person deposits $\rupee 200$ a week for $5$ weeks. Final balance?
4. A bookshop loses $\rupee 80$ on each of $12$ books returned. Total loss?
5. A glacier retreats $3$ m per year. Change over $15$ years?
6. Express "$10$ degrees below freezing" as an integer.
7. A score is $+5$ in one game and $-12$ in the next. Total?
8. Average of $-3, -1, 0, 2, 7$?

Activity

Track the daily temperature in your city for one week. Note rises and falls as positive/negative integers. Find the total change and the average.