Exercises: Operations with natural numbers

Work out 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) $18 - 12 : 3 + 4$

Write the numerical expression corresponding to each statement and work out 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.

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

In a comic shop, a teenager buys $3$ manga books at $8$ € each and $2$ posters at $5$ € each. If he pays with a $50$ € note, how much change should he receive? Express all the calculations in a single line.

Lucas wants to calculate the total cost of buying a notebook for $4,50$ € and $3$ identical pens that cost $1,20$ € each. To do so, he writes the combined operation $4,50 + 3 \cdot 1,20$. When calculating it in his notebook, he first solves the addition: $4,50 + 3 = 7,50$, and then multiplies the result by $1,20$, obtaining $9,00$ €. Identify the mistake made by Lucas, explain which order of operations rule he did not follow, and correctly calculate the total cost he should have paid.

At a local greengrocer's, you buy $3,1$ kg of oranges sold at $1,95$ € per kilogram. Before the scale prints the receipt: a) Mentally estimate the total price by rounding the weight and price to the nearest whole units, and justify whether your estimate will be slightly higher or lower than the actual cost. b) Carry out the multiplication with all its decimals to find the exact amount. c) Check whether the exact result confirms your prediction and assess whether paying that amount makes sense in real life.

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

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

In a video game competition, participants must complete 25 challenges. For each challenge completed, they score $12$ points, but for each mistake, they are penalised $5$ points. If a player decides not to attempt a challenge, they neither gain nor lose points.

Calculate the final score of a contestant who has attempted $22$ challenges, of which $18$ were successful.

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

A farmer has harvested $350$ kg of apples. He sorts them and removes $26$ kg that are unsuitable for sale. He packs the rest into boxes of $12$ kg each. If he sells each box for $18$ €, how much money will he get from selling his whole crop?

To make the costumes for a school play, the art department purchases two pieces of fabric of the same type at $4,25$ € per metre: one measuring $6,7$ metres for the capes and another measuring $3,3$ metres for the suits. Two methods are proposed to calculate the total cost: Method 1, calculate the cost of each piece separately and add them together ($4,25 \cdot 6,7 + 4,25 \cdot 3,3$); Method 2, factor out the common term and add the lengths first ($4,25 \cdot (6,7 + 3,3)$). Solve the operation following both methods step by step, explain which one turns out to be noticeably faster for mental calculation, and mathematically justify why both yield exactly the same result.

Four friends share a car to travel from Sevilla to Granada ($250$ km). The vehicle consumes an average of $6,2$ litres of petrol for every $100$ km travelled, and the price of fuel is $1,65$ € per litre. In addition, they must add a surcharge of $5,90$ € for the toll on the route. If they agree to split all expenses equally: a) Model the situation by writing a single combined mathematical expression (with the necessary parentheses) that allows calculating what each person must pay. b) Link each part of your expression to the key words studied: which operation corresponds to 'calculating the total', which to 'times/product', and which to 'sharing/dividing'? c) Calculate the final result step by step respecting the order of operations and round the final share per person to the nearest euro cent.

Work out the value of the following expression, applying correctly the order of operations and the laws of indices:

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

We are designing the logo for a new mobile app. The logo consists 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 length of the larger square is $12 \text{ mm}$:

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

Apply the distributive property to factorise out the common factor and work out as quickly as possible the value of the following store billing calculation:

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

Design an expression with decimal numbers that simultaneously satisfies these four conditions: 1) It must contain exactly four distinct positive decimal numbers and three different elementary operations (chosen from $+$, $-$, $\cdot$, and $:$). 2) It must include a single pair of parentheses that alters the natural order of operations. 3) When solving the expression respecting the parentheses and the order of operations, its exact final value must be $10$. 4) If those parentheses are completely removed, the resulting expression must give a value strictly less than $5$. Show the expression you have created, work out the calculation step by step in both cases (with and without parentheses) to prove that it satisfies the conditions, and describe the reverse strategy you followed to deduce the numbers and operations.

The two-month water bill in a municipality consists of a fixed service fee of $12,50$ € and a tiered variable consumption: the first $15\text{ m}^3$ consumed are billed at $0,80$ €/$\text{m}^3$, while the cubic metres exceeding that amount are billed at a penalised rate of $1,40$ €/$\text{m}^3$. A family has consumed $26,5\text{ m}^3$ over the two-month period. a) Write a single combined expression that models the total cost of the bill, reflecting the fixed fee, the cost of the basic tier, and the cost of the excess consumption. b) A neighbour claims: 'To avoid doing so many calculations, it is enough to compute the arithmetic mean of the two rates, $(0,80 + 1,40) : 2 = 1,10$ €/$\text{m}^3$, and multiply the total volume ($26,5\text{ m}^3$) by that average rate, then add the fixed fee.' Calculate the result following the neighbour's proposal. c) Compare both results, indicate which one benefits the consumer, and argue mathematically why the neighbour's method makes a conceptual mistake by not weighting the volume of each tier.