Operations with integers

Calculate the following simple additions and subtractions of integers:

1. a) $(+7) + (+5) =$
2. b) $(-8) + (-3) =$
3. c) $(+9) + (-4) =$
4. d) $(-6) + (+10) =$

Solve the following operations by removing the brackets first:

1. a) $12 + (-5) =$
2. b) $8 - (+10) =$
3. c) $(-3) - (-7) =$
4. d) $(-5) + (+5) =$

A Met Éireann weather station in the Wicklow Mountains recorded the following temperature changes over the course of a winter's day:

1. At 8:00 h, the temperature was $-5\ ^\circ C$.

2. At 12:00 h, it had risen by $8\ ^\circ C$.

3. At 18:00 h, it dropped by $4\ ^\circ C$ compared to the previous reading.

4. At 22:00 h, it dropped again by another $6\ ^\circ C$.

Write a single mathematical expression using integers that reflects all these changes, and calculate the final temperature at the end of the day.

Use the rule of signs to solve these multiplications and divisions:

1. a) $(+4) \times (-3) =$
2. b) $(-5) \times (-2) =$
3. c) $(-20) : (+5) =$
4. d) $(-18) : (-3) =$

Apply the distributive property to calculate these expressions in the simplest way possible (factorise out a common factor when necessary):

1. a) $(-7) \times (+12) + (-7) \times (-2)$

1. b) $15 \times (-4) - 15 \times (+6)$

1. c) $(-10) \times (-3) + (-10) \times (+13)$

Calculate the result of these simple combined operations, respecting the order of operations (remember: multiplication and division first, then addition and subtraction):

1. a) $5 + 2 \times (-3) =$
2. b) $(-10) : 2 + 8 =$
3. c) $4 \times 3 - 15 =$

Calculate the result of the following combined operations, paying special attention to the order of operations and the rule of signs:

1. a) $15 - (8 - 12) + (-5) \times 2$

1. b) $(-3) \times [(-2) + (-4) \times (-3)]$

1. c) $24 : (-4) - (-6) : (+3)$

1. d) $(-10) + (-2) \times (-5) + (-20) : 5$

Solve the following combined operations with square brackets. Remember to perform the operations inside the brackets first:

1. a) $18 - [ 2 + (-5) - (4 - 7) ]$

1. b) $[ (-6) \times 2 - (-10) ] : (-2)$

1. c) $30 : [ (-4) + (-2) \times (-3) ]$

1. d) $(-5) \times [ 8 - (12 : (-3)) ]$

To open a mystery box in a maths workshop, you must solve two chained challenges using exclusively the integers $-6$, $+4$, $-2$, and $+3$:

• Challenge 1: Use the four numbers exactly once each, along with the operations of your choice ($+$, $-$, $\times$, $:$) and any parentheses you need, to form a combined expression whose result is exactly equal to $-12$. Write the complete expression and check its result step by step.

• Challenge 2: Without testing random combinations, reason whether it is possible to multiply any three of those four numbers and add the fourth to obtain a strictly positive result. Justify your answer using the sign rules of multiplication and write, if possible, a combination that satisfies it along with its calculation.

Place the necessary parentheses and brackets in the following expression so that the final result is exactly equal to $-5$:

$-15 + 3 \times -4 - 12 : -6 + 1$

Two students, Aoife and Niamh, have to calculate the value of the following expression with integers:

• Aoife applies the standard order of operations: she first performs the two multiplications separately and then adds the obtained results.

• Niamh observes the structure of the calculation and applies the distributive property in reverse (factoring out the common factor): $C = (-15) \times (74 + 26)$.

1. a) Calculate the result following in detail Aoife's method and Niamh's method. Verify that both approaches reach the same solution.
2. b) Compare the two methods and justify which of them is more efficient for mental calculation and why it reduces the risk of making mistakes with signs and intermediate operations.
3. c) Design a new combined operation consisting of the addition or subtraction of three products of integers where applying Niamh's strategy allows it to be solved mentally in a few seconds without the need for long calculations on paper. Show your expression and its justified resolution.

Find the integer that simultaneously satisfies these three conditions:

1. Its modulus is greater than $5$ and less than $10$.

2. When multiplied by $-3$, the result is a positive number.

3. If you add its opposite to it, the result is $0$.

Is there more than one possible solution? Justify your answer.

Find the unknown integer $x$ in each of the following equalities:

1. a) $x \times (-4 + 9) = -25$

1. b) $(-30) : x + (-12) = -18$

1. c) $2 \times |x| = 14$ (Find all possible solutions for $x$).

Three unknown integers, $a$, $b$, and $c$, simultaneously satisfy the following three conditions:

1. Their product is strictly negative: $a \times b \times c < 0$.

2. The division $a : b$ is exact and results in a positive integer: $a : b > 0$.

3. The sum of two of them is zero: $a + c = 0$, also knowing that $a \neq 0$.

1. a) Logically and reasonably deduce the sign (positive or negative) of each of the three numbers ($a$, $b$, and $c$). Justify each step based on the concept of opposite numbers and the sign rules of multiplication and division.
2. b) If it is known that $|b| = 2$ and that the product of the three numbers is $a \times b \times c = -72$, determine the exact numerical values of $a$, $b$, and $c$.
3. c) With the values found in the previous section, calculate the result of the following combined operation respecting the order of operations and the sign rules:

You are on floor $2$ of a shopping centre. You go up $3$ floors to get to the food court and then go down $6$ floors to reach the car park. Which floor is the car park on? State each movement using an integer.

On the bank statement of a comic book shop, the following transactions appear:

• Starting balance: $+\text{€}\ 450$

• Payment to supplier: $-\text{€}\ 620$

• Sale of collectible figures: $+\text{€}\ 185$

• Refund for a defective item: $-\text{€}\ 45$

• Income from subscriptions: $+\text{€}\ 110$

Calculate the final balance of the account. If the bank charges a fee of $\text{€}\ 15$ if the account goes overdrawn (balance less than 0), will the shop owner have to pay it?

A caving team is exploring a deep chasm. They start at the entrance (level $0$ metres). They make the following moves consecutively:

1. They descend $45$ metres.

2. They climb up $12$ metres to set up a base camp.

3. They descend three times as far as they have just climbed.

4. Finally, they descend another $18$ metres to the bottom.

Write a single expression using integers reflecting the whole process and calculate the depth of the bottom of the chasm.