We welcome you to the world of integers, where we will explore a series of problems that will challenge your mathematical skills and your ability to solve real-world situations using basic operations. In this series of problems, we will delve into scenarios ranging from financial calculations to physical concepts, through everyday situations that require a solid understanding of operations with integers.

1

Augustus, Roman emperor, was born in 63 BC and died in 14 AD. How many years did he live?

Solution

1. We find the years he lived before Christ:

2. We find the years he lived after Christ:

3. We add both results:

years

2

A pump extracts oil from a well at a depth of 3,200 feet and raises it to a tank located 158 feet above ground. What level does the oil overcome?

Solution

1. The level that the oil overcomes is the sum of the height and the depth.

2. Considering ground level as height zero, the height is:

3. The depth is:

Thus, the level that the oil overcomes is the sum of the height and the depth:

Another way is to consider the depth negative and the height positive; then the level sought is the difference of the extremes, that is:

3

A worm is at the bottom of a well. During the day it climbs 6.6 feet and during the night it descends 3.3 feet. What height has it climbed after three days and two nights?

Solution

1. In three days the worm climbs:

2. In two nights the worm will have descended:

3. The worm has climbed in total:

4

Juan saved $12 on Monday; on Tuesday he saved half of what he saved on Monday, and on Wednesday he saved a third of what he saved on Tuesday. How much do Juan's savings amount to?

Solution

1. On Tuesday he saved half of what he saved on Monday, that is:

dollars

2. On Wednesday he saved a third of what he saved on Tuesday:

dollars

3. Juan's savings amount to:

dollars

5

What temperature difference does a person endure who goes from the vegetable storage room, which is at 39°F, to the frozen fish room, which is at -0.4°F? What if they went from the fish room to the vegetable room?

Solution

1. The temperature difference from going from the vegetable storage room to the frozen fish room is:

The negative sign indicates a decrease in temperature.

2. The temperature difference from going from the frozen fish room to the vegetable storage room is:

The positive sign indicates an increase in temperature.

6

Air temperature decreases as you ascend into the atmosphere, at a rate of 16°F every 1,000 feet. At what altitude is an airplane flying if the air temperature is -114°F and it took off at 32°F?

Solution

1. First we find the total temperature drop:

2. We divide the temperature drop by the rate:

3. To find the altitude, we recall that each decrease of 16°F corresponds to an altitude of 1,000 feet:

Thus, the airplane is flying at an altitude of approximately 9,000 feet.

7

Ana has a basket of apples. If she takes out 17 apples and forms groups of three apples, how many additional apples must she take from her basket so that all the apple groups are complete?

Solution

1. First we divide 17 into groups of three using division:

So Ana forms 5 groups of three apples and has two apples left over.

2. Ana must take one additional apple from her basket to complete groups of three.

8

One painter takes one day to paint a house. A second painter takes two days to paint a house. If both work together, how many houses will they paint in six days?

Solution

1. The first painter paints 6 houses in 6 days.

2. The second painter paints 3 houses in 6 days.

3. Together, the two painters paint 9 houses in 6 days.

9

A tank has two pipes that pour 5.3 gallons and 7.9 gallons of water per minute. There is a drain through which 4.0 gallons per minute exits. How many gallons of water will be in the tank after 10 minutes?

Solution

1. The amount of water entering the tank through the first pipe in 10 minutes is:

2. The amount of water entering the tank through the second pipe in 10 minutes is:

3. The amount of water leaving the tank in 10 minutes is:

4. The total amount of water in the tank after 10 minutes is:

10

A tank has 211 gallons of water. Through the top, a pipe pours 6.6 gallons per minute into the tank, and through the bottom, another pipe releases 7.9 gallons per minute. How many gallons of water will be in the tank after 15 minutes of operation?

Solution

1. The amount of water entering the tank in 15 minutes is:

2. The amount of water leaving the tank in 15 minutes is:

3. The total amount of water in the tank after 15 minutes is:

11

Juan bought a television for $300 and later sold it for $150 more than what he paid. What was his profit or loss?

Solution

Money spent on the purchase: $300.

Money obtained from selling: $300 + $150 = $450.

Profit: $450 - $300 = $150.

12

The temperature in a town dropped 14°F during the night and then rose 9°F during the day. What was the net change in temperature?

Solution

It dropped 14°F; it rose 9°F. Therefore, there is a net decrease of 14 - 9 = 5°F.

13

A car travels 25 miles north and then 15.5 miles south. What is the total distance traveled and in what direction is the car relative to its starting point?

Solution

The car moved a total of 25 mi - 15.5 mi = 9.5 miles north.

14

Pablo owes his friend $500 and then pays $200 of that debt. What is his total debt now?

Solution

His debt is $500 - $200 = $300.

15

A ship moves 7.5 miles east and then 5 miles west. In what direction and at what distance is it from the starting point?

Solution

It moved a total of 7.5 mi - 5 mi = 2.5 miles east.

16

If Pedro's age is 15 years more than his younger brother's age and his younger brother is 10 years old, how old is Pedro?

Solution

Pedro's age is 10 + 15 = 25 years.

17

Find three consecutive integers whose sum is 21.

Solution

Suppose is some number that satisfies:

Note that these are 3 consecutive numbers. Simplifying the expression we get:

That is, 6 + 7 + 8 = 21.

18

If we multiply an integer by 4 and then add 12, we get 44. What is the integer?

Solution

Let be such a number. Then:

 

19

A scale model of a building is reduced to 1/4 of its original size. If the original was 52.5 feet tall, how tall is the model?

Solution

We reduce to 1/4 of 52.5, that is:

Therefore, we reduce to 13.1 feet.

20

A cyclist moves at 12.4 miles per hour for 3 hours and then at 9.3 miles per hour for 2 hours. What is the total distance traveled by the cyclist?

Solution

In the first 3 hours, he travels:

while in the next 2 hours he travels:

so in total, he travels 37.2 mi + 18.6 mi = 55.8 miles.

Summarize with AI:

Did you like this article? Rate it!

5.00 (2 Note(n))
Loading...

Agostina Babbo

Agostina Babbo is an English and Italian to Spanish translator and writer, specializing in product localization, legal content for tech, and team sports—particularly handball and e-sports. With a degree in Public Translation from the University of Buenos Aires and a Master's in Translation and New Technologies from ISTRAD/Universidad de Madrid, she brings both linguistic expertise and technical insight to her work.