Operations with natural numbers
At a comic book shop you buy 3 manga volumes at 160 pesos each and 2 posters at 100 pesos each, and you pay with a 1,000-peso bill. How much change do you get back? You can solve it in a single line if you know how to translate the problem into numbers and follow the correct order of operations.
From a word problem to an operation
First you have to turn the words into operations. These words can help you:
Write the numerical expression corresponding to each statement and work out its value: a) Three times the sum of eight and five. b) Twice twenty minus the quotient of…
Write the numerical expression corresponding to each statement and work out its value:
- a) Three times the sum of eight and five.
- b) Twice twenty minus the quotient of twelve divided by four.
- c) The difference between the product of six and three, and the number seven.
In a comic shop, a teenager buys $3$ manga books at $160$ pesos each and $2$ posters at $100$ pesos each. If he pays with a $1\,000$ pesos note, how much change should…
In a comic shop, a teenager buys $3$ manga books at $160$ pesos each and $2$ posters at $100$ pesos each. If he pays with a $1\,000$ pesos note, how much change should he receive? Express all the calculations in a single line.
The order of operations in practice
An expression with several operations is always solved in this order: first parentheses and brackets (starting with the innermost ones), then powers and roots, then multiplications and divisions, and finally additions and subtractions. That way everyone gets the same result.
When there are brackets, you first solve what is inside the innermost parentheses. For example, $[2^5 : (2^2 \cdot 4) + 25]$: you calculate $2^2 \cdot 4 = 16$, then $32 : 16 = 2$, and finally $2 + 25 = 27$.
Properties of powers for calculating faster
Sometimes it is better to use the properties of powers instead of calculating each one separately. Here we only use them; you will study them in detail in the section on powers.
Work out the result of the following combined operations, paying attention to the order of operations: a) $25 + 5 \times 4 - 10$ b) $36 : 6 + 2 \times 7$ c)…
Work out the result of the following combined operations, paying attention to the order of operations:
- a) $25 + 5 \times 4 - 10$
- b) $36 : 6 + 2 \times 7$
- c) $18 - 12 : 3 + 4$
Work out the result of the following combined operation step by step, respecting the order of operations: $15 \times (13 - 3^2) + 4^2 : \sqrt{64} - 2 \times 5$
Work out the result of the following combined operation step by step, respecting the order of operations:
$15 \times (13 - 3^2) + 4^2 : \sqrt{64} - 2 \times 5$
Work out the value of the following expression, applying correctly the order of operations and the laws of indices: $$ \sqrt{10^2 - 8^2} + [2^5 : (2^2 \times \sqrt{16})…
Work out the value of the following expression, applying correctly the order of operations and the laws of indices:
$$ \sqrt{10^2 - 8^2} + [2^5 : (2^2 \times \sqrt{16}) + 5^2] - (3^3 - 5^2)^2 $$Remember to solve the innermost brackets first, followed by powers and roots, then multiplications and divisions, and finally additions and subtractions.
Problems with combined operations
In a multi-step problem, it helps to write everything as a single expression and follow the order of operations. That also lets you check your reasoning.
Areas and perimeters of figures made of squares
An area is also calculated with combined operations. If a square has a side of $12$ mm, its area is $12^2 = 144$ mm². If the total area of two squares joined together is $169$ mm², the other one measures $169 - 144 = 25$ mm², and its side is $\sqrt{25} = 5$ mm.
The perimeter of a figure is the length of its outer outline. If you join two squares, the stretch where they are joined is no longer part of the outline: you have to remove it. One way to find it is to add the perimeters of the two squares and subtract the joined stretch twice (once for each square).
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…
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 video game competition, participants must complete 25 challenges. For each challenge completed, they score $12$ points, but for each mistake, they are penalised $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 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…
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 $360$ pesos, how much money will he get from selling his whole crop?
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…
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}$:
- a) What is the area of the small square?
- b) What is the total perimeter of the resulting shape (the outer outline)?
Properties of operations and common factor
Commutative and associative
When adding and multiplying natural numbers, you can change the order and the grouping without changing the result.
Distributive property and common factor
Since $125 = 5^3$, if that result is shared into bags of $5^3$ items, the number of bags is $(5^3 \cdot 100) : 5^3 = 100$.
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 $300$ pesos and the remaining tickets for…
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 $300$ pesos and the remaining tickets for $180$ pesos. If all seats have been sold, what was the total revenue for the screen?
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: $$ 125…
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:
$$ 125 \times 48 + 125 \times 64 - 125 \times 12 $$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.
Back to the beginning: we translated the problem into a single operation: $1{,}000 - (3 \cdot 160 + 2 \cdot 100) = 1{,}000 - (480 + 200) = 1{,}000 - 680 = 320$. You get 320 pesos back in change.