Operations with Natural Numbers

Write the numerical expression that corresponds to each statement and calculate its value:

1. a) Three times the sum of eight and five.

1. b) Twice twenty minus the quotient of twelve divided by four.

1. c) The difference between the product of six and three and the number seven.

A group of four friends wants to buy a subscription to a video game platform that costs $\$48$. If they already have $\$12$ saved and want to pay the rest equally, write the mathematical expression that allows calculating how much each one must contribute and find the result.

In a comic book store, a teenager buys $3$ mangas at $\$8$ each and $2$ posters at $\$5$ each. If they pay with a $\$50$ bill, what is the amount of change they should receive? Express all operations on a single line.

Calculate the result of the following combined operations paying attention to the order of operations:

1. a) $25 + 5 \cdot 4 - 10$

1. b) $36 : 6 + 2 \cdot 7$

1. c) $18 - 12 : 3 + 4$

Solve the following expressions taking into account the importance of parentheses:

1. a) $100 - (20 + 30 + 10)$

1. b) $(15 - 5) \cdot (8 - 6)$

1. c) $5 \cdot (12 : 4) + (9 - 3)$

Calculate the result of the following combined operation step by step, respecting the order of operations:

$15 \cdot (13 - 3^2) + 4^2 : \sqrt{64} - 2 \cdot 5$

Calculate the result of the following expression correctly applying the order of operations and the properties of powers:

$15 + 2 \cdot (3^3 - \sqrt{144}) : (2^2 + 1) + 5^2$

Apply the order of operations rules to solve the following combined operation, paying attention to parentheses and roots:

$(\sqrt{64} + 2^3) \cdot [12 - (10 - 4)] : 2^2$

Calculate the value of the following expression correctly applying the order of operations and the properties of powers:

Remember to solve the innermost parentheses first, followed by powers and roots, then multiplications and divisions, and finally additions and subtractions.

Look at the following grid that represents a storage area in a logistics warehouse. Calculate the total number of packages using a single multiplication and addition operation, knowing that each square represents a box.

An e-sports competition is to be organized with a certain number of participants. If we organize them at rectangular tables as shown in the drawing, how many players will there be in total if we have $12$ identical tables?

In a logistics warehouse, pallets of soft drinks are being organized. There are $8$ towers of boxes, and each tower has $8$ levels. On each level there are exactly $8$ boxes of soft drinks.

1. a) Express the total number of boxes as a power.
2. b) If each box contains $12$ bottles and they are to be distributed equally among $4$ trucks, how many bottles will each truck carry?

A farmer has harvested $350$ pounds of apples. They sort them and remove $26$ pounds that are not suitable for sale. The rest are packed into boxes of $12$ pounds each. If they sell each box for $\$18$, how much money will they get from the sale of their entire harvest?

In a video game competition, participants must overcome 25 challenges. For each challenge overcome, $12$ points are obtained, but for each failure, there is a penalty of $5$ points. If a player decides not to attempt a challenge, they neither add nor subtract points.

Calculate the final score of a contestant who has completed $22$ challenges, of which $18$ have been successful.

A stationery store prepares sets of school supplies for the start of the academic year. Each set contains $4$ spiral notebooks and $3$ ring binders. The owner wants to prepare $15$ identical sets.

1. a) Write a mathematical expression that uses the distributive property to calculate the total number of items needed.
2. b) Calculate how many items there will be in total between notebooks and binders.

A movie theater has a screen with $12$ rows of $18$ seats each. For a premiere, the tickets for the first $4$ rows (VIP area) are sold at $\$15$ and the rest of the tickets at $\$9$. If all the seats have been sold, what has been the total revenue for the theater?

Apply the distributive property to extract the common factor and calculate the value of the following shop billing operation as quickly as possible:

If the total result is to be distributed into bags containing $5^3$ items each, how many bags will be filled? Express the process using powers.

Decompose the following numbers in polynomial form using powers of base $10$, as shown in the example:

Example: $3\,427 = 3 \cdot 10^3 + 4 \cdot 10^2 + 2 \cdot 10^1 + 7 \cdot 10^0$

1. a) $5\,832$

1. b) $12\,045$

1. c) $906$

We are designing the logo for a new mobile app. The logo is made up of two perfect squares joined so that one full side of the small square rests against a side of the large square. The total area of the logo is $169 \text{ mm}^2$. If the side of the larger square measures $12 \text{ mm}$:

1. a) What is the area of the small square?
2. b) What is the total perimeter of the resulting figure (the outer outline)?

Triangular numbers ($T_n$) are formed following the geometric pattern in the figure. The fourth triangular number is $T_4 = 10$.

1. a) Knowing that the formula to find any triangular number is $T_n = \frac{n \cdot (n + 1)}{2}$, calculate the triangular number in position $15$ ($T_{15}$).
2. b) There is a property that says that the sum of two consecutive triangular numbers is always a perfect square ($T_{n-1} + T_n = n^2$). Prove this by adding $T_9$ and $T_{10}$ and calculate the square root of the result.

A coastal lighthouse emits a white light every $12$ seconds, a red light every $15$ seconds and a blue light every $18$ seconds. At $21:00$ hours the three lights have coincided for the first time in the night.

1. a) What is the minimum number of seconds that must pass for the three lights to coincide again?
2. b) How many times will the three lights coincide in total from $21:00$ to $21:10$ hours (inclusive)?

In a robotics competition, the robots must cover a track of $540 \text{ in}$. Robot A makes constant jumps of $12 \text{ in}$, robot B makes jumps of $15 \text{ in}$ and robot C of $18 \text{ in}$.

1. a) At what points on the track (measured in inches from the start) will the footprints of the three robots coincide at the same time?
2. b) If the energy cost of each jump is $5$ units for any robot, which of the three robots consumes the least energy to complete the entire track? Justify your answer with calculations.

A logistics company has to distribute $252$ cartons of orange juice and $324$ cartons of pineapple juice into identical boxes, so that they have the same number of cartons and this is as large as possible, without mixing flavors in the same box.

1. a) How many cartons will there be in each box?
2. b) Due to an error, $12$ cartons of pineapple juice are broken before being packed. Will they be able to continue using the same box size for the remaining pineapple cartons so that the boxes are still full? Justify your answer.

A hacker has left an encrypted message where a four-digit number $3x5y$ is the key. To decrypt it, we know that:

• The number is divisible by $4$.

• The number is divisible by $9$.

Find all possible values for the digits $x$ and $y$ that meet both conditions. What is the largest possible number that can be the key?