Exercises: Integers

Evaluate the following combined operation respecting the order of operations:

$5 - [ 3 - ( -4 + 7 ) \cdot 2 ] + (-8) : 4$

Work out the numerical value of the following expression containing absolute values:

$|-15 + 8| - |3 - 9| + |-2|$

Determine the sign and work out the exact value of the following powers of integers:

• $(-2)^4$

• $(-3)^3$

• $-5^2$

• $(-1)^{15}$

An industrial freezer is at a temperature of $-18$ °C. If due to a power failure the temperature rises by $5$ °C and, after repairing it, the motors manage to lower it by $12$ °C, what is the final temperature of the freezer?

Lucas tried to solve the following combined operation on the board: $15 - (7 - 12) + (-3)^2$. His working was:

$15 - (7 - 12) + (-3)^2 = 15 - 7 - 12 - 9 = 8 - 12 - 9 = -13$

However, his teacher told him that the result is incorrect because he made two serious conceptual errors regarding the rules of integers.

1. a) Identify the two mistakes made by Lucas and explain with reasons why they contradict the rules of signs in front of parentheses and the calculation of powers with a negative base.
2. b) Solve the operation correctly step by step.

The State Meteorological Agency (AEMET) recorded a temperature of $-6$ °C at 07:00 at the Pradollano weather station (Sierra Nevada, Granada). Throughout the day, the following consecutive temperature changes were recorded: at midday the temperature rose by $9$ °C, in the mid-afternoon it dropped by $4$ °C due to a moist air mass, at dusk it fell sharply by $7$ °C, and, during the night, the wind made it drop an additional $2$ °C.

1. a) Write a single arithmetic expression using integers that represents all the temperature changes experienced at the station.
2. b) Calculate the final temperature shown on the thermometer at the end of the night.
3. c) If the winter resort activates the severe frost warning when the temperature drops below $-8$ °C, was this warning activated by the end of the night? Justify your answer by comparing the corresponding integers.

The balance of a bank account at the beginning of the month is $-150$ €. Over the course of the week, the following transactions take place: a cash deposit of $400$ €, an electricity bill payment of $85$ €, an outgoing bank transfer of $230$ € and a maintenance fee charge of $12$ €. Write down the combined calculation and determine the final balance of the account.

An underwater drone carries out an ocean exploration. It begins its dive by descending $45$ m from the surface. Then it descends a further $120$ m to investigate a trench. After taking samples, it ascends $85$ m and stops to photograph a coral. Finally, it descends $30$ m more to return to the sea floor. At what depth is it relative to sea level?

Work out the following combined calculation, paying special attention to square brackets and powers with negative bases:

$12 - [(-2)^3 + 5 \cdot (-3)] : (-1) + (-4)^2$

In a basketball tournament, the standings are decided by the point difference (difference between points scored and points conceded). In the first four matches, a team's results were: $75-82$, $90-85$, $68-77$ and $85-80$. Express each match's point difference as an integer and work out the team's total point difference at the end of the fourth round.

A family from Zaragoza checks the bank transactions in their checking account over the last month. The initial balance was $+380$ €. Throughout the month, the following entries are recorded: a salary deposit of $+1420$ €, a rent payment of $-690$ €, an electricity bill of $-185$ €, accumulated food expenses of $-310$ €, a tax refund credit of $+195$ €, and an urgent boiler repair for $-820$ €.

1. a) Before calculating the exact result, make an estimate by rounding each amount to the nearest hundred. Determine, from your estimate, whether the family will end the month with a positive balance or in the red (negative balance).
2. b) Set up the addition of integers with the exact values and calculate the final account balance.
3. c) Compare the exact balance obtained with your previous estimate and assess whether estimating before calculating was useful and accurate in anticipating the family's financial situation.

To calculate the value of the expression $-3 \cdot [4 - (-5 + 9) \cdot 2 - 7]$, two 4th-year ESO students propose different strategies:

• Marina's strategy: Work from the inside out; that is, solve the inner parentheses $(-5 + 9)$, carry out the multiplication by $2$, perform the additions and subtractions inside the brackets, and finally multiply the result obtained by $-3$.

• Sergio's strategy: Apply the distributive property by directly eliminating the brackets, multiplying each of the inner terms by $-3$ before solving the inside.

1. a) Solve the operation strictly following Marina's strategy.
2. b) Solve the operation following Sergio's strategy, paying special attention to the sign of each product.
3. c) Explain which of the two strategies is more convenient and less prone to sign errors in this specific case, justifying your choice based on the order of operations.

The city of Rome was traditionally founded in $753$ BC and the Western Roman Empire fell in $476$ AD. Given that in the Gregorian historical calendar there is no year $0$ (it goes directly from $1$ BC to $1$ AD), write an operation using integers to work out exactly how many years passed between both historical events.

A computer program modifies a counter $C$ (whose initial value is $C = 0$) using a loop that repeats $100$ times (iterations numbered from $1$ to $100$). On even iterations, the program adds $5$ to the counter; on odd iterations, it subtracts $8$. Write an expression using integers that models the process and work out the exact value of the counter $C$ at the end of the $100$ iterations.

Design a combined arithmetic expression that simultaneously satisfies all the following requirements:

1. It must use exactly four distinct integers, of which at least two must be strictly negative.

2. It must contain a power with a negative base and an even exponent, of the form $(-a)^2$, and an absolute value $|b - c|$ where the subtraction inside yields a negative result before applying the absolute value bars.

3. It must include at least one multiplication and one subtraction that lie outside the absolute value and the parentheses of the power.

4. When solving the expression following the order of operations, the final result must be exactly $-16$.

Write the complete expression you have designed and show, by solving it step by step, that it meets each of the required conditions.

In an interdisciplinary History and Mathematics project, three key archaeological milestones of the Iberian Peninsula are investigated:

1. The end of the Celtiberian siege of Numancia (Soria) occurred in the year $133$ BC.

2. The official foundation of the Roman colony of Caesaraugusta (present-day Zaragoza) took place in the year $14$ BC.

3. The inauguration of the Aqueduct of Segovia during the imperial period is dated to the year $112$ AD.

Bearing in mind that in conventional historical dating there is no year $0$ (the year $1$ BC is followed immediately by the year $1$ AD):

1. a) Model using integers and calculate how many years elapsed between the fall of Numancia and the foundation of Caesaraugusta.
2. b) Model and calculate how many years elapsed between the foundation of Caesaraugusta and the inauguration of the Aqueduct of Segovia, mathematically justifying why it is necessary to subtract $1$ when calculating the time span between BC and AD dates.
3. c) A tourist brochure claims that between the fall of Numancia and the inauguration of the Aqueduct of Segovia exactly $133 + 112 = 245$ years elapsed. Explain why this calculation is incorrect, write the appropriate mathematical operation, and find the exact number of years that elapsed.