Exercises: Operations with decimal numbers

In a mobile phone accessory shop, Marc wants to buy a case costing $12,45$ € and a screen protector costing $8,90$ €. If Marc pays with a $50$ € note, how much change will he receive at the till?

Work out these calculations mentally using powers of ten. Remember to move the decimal point to the correct side depending on whether you multiply or divide:

1. a) $34,57 \cdot 10$
2. b) $0,23 \cdot 100$
3. c) $458,2 : 10$
4. d) $12,5 : 1000$

Marta went to the greengrocer's and bought $2,5$ kg of apples at $1,80$ € per kilogram. On arriving home, she notices that one of the apples is bad and decides to return it. If the apple weighed $0,25$ kg, how much money should Marta be refunded?

To make a recipe, we need to divide a $1,5$ kg bag of flour into $6$ equal portions to make muffins. How many kilograms of flour will each portion weigh? Express your answer in grams as well.

Work out the following combined operations, respecting the order of operations (brackets, multiplications/divisions, and finally additions/subtractions):

1. a) $4,5 + 2,1 \cdot 3$
2. b) $(12,8 - 3,4) : 2$
3. c) $5 \cdot 0,25 + 1,75$

A group of $4$ friends goes to the cinema. The total for tickets and popcorn came to $45,80$ €. If they decide to split the cost equally, how much money must each person pay? If one of them only has a $10$ € note and a $2$ € coin, will they have enough money?

Copy and complete the following statements by writing the missing decimal number to make each calculation correct:

1. a) $15,45 + \dots = 20$
2. b) $7,2 - \dots = 4,55$
3. c) $\dots \cdot 3 = 12,9$

An athlete is training on a circular track. Each lap of the track is $0,45$ km long. If today she completed a total of $12$ laps, how many kilometres did she cover in total? If her goal was to run $6,3$ km, how many more laps should she have completed?

Sofía has a wooden strip of $14,2\text{ m}$ and needs to cut a piece of $3,75\text{ m}$ to build a shelf. To calculate how much wood she has left over, she sets up the subtraction $14,2 - 3,75$ vertically and claims that the result is $11,55\text{ m}$, reasoning as follows: "in the hundredths column, since there is nothing at the top, I bring down the $5$ directly; in the tenths I subtract the smaller digit from the larger one, $7 - 2 = 5$; and in the units I subtract $14 - 3 = 11$".

1. a) Identify and explain the two conceptual mistakes made by Sofía related to the place value of the digits and the subtraction operation.
2. b) Explain why it is essential to add zeros to the right of the decimal part of the minuend before calculating.
3. c) Set up the column operation correctly and find the exact length of the strip that Sofía has left.

You go to your local grocery store with a single $15\text{ €}$ banknote and put the following items in your basket:

• $1$ pack of coffee for $3,95\text{ €}$,

• $2$ cartons of milk at $1,05\text{ €}$ each,

• $1$ loaf of artisanal bread for $0,98\text{ €}$, and

• $3$ packs of biscuits at $1,99\text{ €}$ each.

1. a) Before doing any written calculations with decimals, make a mental estimate by rounding the price of each item to the nearest whole euro. Using this estimate, reason whether you will have enough money with the $15\text{ €}$ banknote to pay for the purchase.
2. b) Now perform the exact operations in columns step by step to calculate the total amount shown on the cash register.
3. c) Check whether your initial estimate was correct, calculate exactly how much money you have left over or are missing, and reflect on situations in everyday life where it is useful to estimate before calculating.

Three friends decide to have an afternoon snack together. They buy a pack of soft drinks for $4,85$ €, a family-size bag of crisps for $2,30$ € and a box of small cakes for $5,45$ €. If they pay with a $20$ € note, how much money will they have left over to buy sweets afterwards?

A spool of filament for a 3D printer weighs exactly $1,25$ kg. If $0,015$ kg of plastic is used to make a small part, how many complete parts can be made from a single spool? How much filament will be left over at the end?

Valentina wants to download three video game files on her console. The first file is $4,35$ GB, the second is double the first, and the third is $1,2$ GB less than the second. If her connection downloads at a constant speed of $0,05$ GB per second, how many seconds will it take to complete the download of all three files?

Calculate the result of the following combined operations, respecting the order of operations:

1. a) $12,5 + 3,8 \cdot (6,4 - 2,15)$

1. b) $(15,4 + 8,6) : 2,5 - 1,02 \cdot 4$

A local council wants to fence a rectangular park measuring $25,4$ metres long and $12,75$ metres wide. The price of the fence is $15,20$ € per metre. If the total available budget is $1.200$ €, do they have enough money to enclose the entire park? Justify your answer with the necessary calculations.

In a "Back to School" offer, a pack of $5$ notebooks costs $8,25$ €, whereas if you buy them individually, each one costs $1,80$ €.

What is the total saving if we decide to buy $15$ notebooks using the pack offer instead of buying them separately?

A marathon runner has completed a stage of $15,6$ km in an hour and a half ($1,5$ hours).

1. a) What was their average speed in km/h?
2. b) If they maintain the same speed, how many kilometres will they cover in $2,25$ hours?

A courier service charges a fixed fee of $4,50$ € per delivery, plus $0,85$ € for each kilogram of the parcel's weight. If we want to send a parcel weighing $3,4$ kg and apply a discount voucher of $1,25$ € to the final price, how much will the delivery cost? Write a single numerical expression for the calculation and solve it.

On a receipt from a fruit shop, part of it has been erased due to a moisture stain. The only clearly legible data are the following:

• Apples: $3,5\text{ kg}$ at an unknown price per kilogram.

• Bananas: $1,2\text{ kg}$ at $1,85\text{ €/kg}$.

• Payment handed over at the till: a $20\text{ €}$ banknote.

• Exact change received: $11,48\text{ €}$.

1. a) Use the method of inverse operations: first calculate the total cost of the purchase and, subtracting the amount corresponding to the bananas, determine how much the apples cost in total.
2. b) From the total cost of the apples, set up and solve the decimal division needed to find the exact price of one kilogram of apples.
3. c) Reconstruct the full calculation forward to check that all operations and the change returned are correct.

At an AEMET weather station located in the region of Cazorla (Jaén), the following precipitation data were recorded over four consecutive days:

• Monday: $16,5\text{ l/m}^2$ were collected.

• Tuesday: it rained $4,2\text{ l/m}^2$ less than on Monday.

• Wednesday: the rainfall was equal to $1,2$ times the amount recorded on Tuesday plus an additional storm contribution of $0,84\text{ l/m}^2$.

• Thursday: exactly the arithmetic mean (the average) of the rainfall over the previous three days was recorded.

1. a) Model the situation by stating and calculating step by step the exact amount of rain collected on each of the four days (clearly show the placement of the decimal comma in each calculation).
2. b) Calculate the total accumulated precipitation over the four days.
3. c) If each litre per square metre recorded in that basin directly contributes a volume of $10\,000\text{ m}^3$ of water to a nearby reservoir, calculate how many cubic metres of water entered the reservoir in total during those four days by applying the rule for multiplying by powers of $10$.

Calculate the result of the following mixed operation, strictly respecting the order of operations and working with all decimal places obtained:

Once the final result is obtained, approximate it by rounding to the nearest hundredth and by truncating to the nearest tenth.

In a technology shop, a video game costs $59,95 €$. During "Black Friday" week, they offer a discount of a quarter of its value. If a teenager decides to buy it and hands over a $50 €$ note, calculate:

1. a) The final price of the video game after applying the discount (round to the nearest euro cent if necessary).
2. b) Will they have enough money? If they are short, state how much. If they have money left over, state the exact change they will receive.

A farmer has a water tank with a total capacity of $1\,250,5$ litres. Due to a small crack, the tank loses $0,75$ litres every hour. At the same time, a water pump pours $12,4$ litres into the tank every $15$ minutes. If the tank currently contains $840,25$ litres:

How many litres will be in the tank after exactly $4$ hours and a half? Express the result to two decimal places.

Look at the rectangle in the figure, with base $b = 12,45 \text{ cm}$ and height $h = 8,2 \text{ cm}$.

Calculate its area and its perimeter. Then, imagine that we reduce both the base and the height to half. What will the new area be? Is the resulting area half of the original or has it changed in some other way? Justify your answer by carrying out the necessary operations.

Three friends want to buy a subscription to a streaming platform that costs $17,99 €$ per month. They decide to split the cost as follows: Juan will pay half, Marta will pay one third of the remainder, and Lucía will pay whatever is left.

Calculate the exact amount (with all its decimal places) that each person must contribute. If the bank only allows payment to two decimal places, who do you think should absorb the rounding of the extra cent so that the sum is exactly $17,99 €$?

A Formula 1 car took $1 \text{ minute}$ and $24,352 \text{ seconds}$ to complete one lap of the circuit. If its main rival took $84,12 \text{ seconds}$ to cover the same distance:

1. a) Which of the two was faster?
2. b) Calculate the time difference between them expressed in seconds.
3. c) If the first car maintains its pace, how long will it take to complete $15$ full laps? Express the result in minutes and seconds.

We are cooking a recipe that requires $0,250 \text{ kg}$ of flour to make $6$ muffins. If we want to invite the whole class ($28$ students) and we want each of them to eat $2$ muffins:

1. a) How much flour will we need in total? Express the result in kilograms.
2. b) If we buy flour packets of $0,75 \text{ kg}$, how many packets will we need to buy at least? How much flour will we have left over exactly?

Alex's mobile phone plan has a fixed monthly fee of $22,50 €$. If he exceeds the included gigabytes, each extra GB costs $0,50 €$, and each minute of international calls costs $0,12 €$. This month he has made $45$ minutes of international calls and the cost of the extra GB was exactly one fifth of the fixed fee:

1. a) How many extra GB has he used?
2. b) What is the total bill before tax?
3. c) If $21\%$ VAT must be added to the total (multiply by $0,21$ and add to the total), what will be the final price of the bill? (Round the final result to the nearest euro cent).

For a telecommunications installation in an educational centre, a $45\text{ m}$ long network cable reel is available. Identical sections of $0,75\text{ m}$ must be cut, and each section will be charged at a price of $2,40\text{ €}$. Two students propose different methods to calculate the total amount of money that will be obtained:

• Alba's method: Works directly in metres and euros with decimals. First divides $45 : 0,75$ applying the rule of multiplying both dividend and divisor by $100$ ($4500 : 75$) so that the divisor is an integer, and then multiplies the quotient by $2,40$.

• Bruno's method: Converts all quantities to integers beforehand by changing units: converts the $45\text{ m}$ to centimetres ($4500\text{ cm}$), the $0,75\text{ m}$ to centimetres ($75\text{ cm}$), and the unit price to cents ($240\text{ cents}$). Solves the operations with integers and in the end converts the result to euros by dividing by $100$.

1. a) Fully work out both methods and check that both yield exactly the same total amount.
2. b) Justify mathematically why the rule of "removing decimals from the divisor by multiplying by powers of $10$" is fundamentally equivalent to the change of units that Bruno performs.
3. c) Now suppose that the required sections had to measure $0,7\text{ m}$. When performing the equivalent integer division $450 : 7$, a quotient of $64$ and a remainder of $2$ are obtained. Reasonably explain what the actual length of leftover cable from the reel is ($2\text{ m}$, $0,2\text{ m}$, or $0,02\text{ m}$) and what precaution you should always take when interpreting the remainder of a division with decimals.

A residential monthly electricity bill in Spain details the following regulated items:

• Fixed power charge: $0,115\text{ €}$ per day is charged for each kilowatt ($\text{kW}$) contracted. The household has contracted $4,6\text{ kW}$ and the billing period is $30\text{ days}$.

• Energy consumption charge: charged at a price of $0,148\text{ €}$ for each kilowatt-hour ($\text{kWh}$) consumed.

• Electricity tax: calculated by multiplying the subtotal (the sum of the power charge plus the consumption charge) by the decimal factor $0,0511$.

1. a) Model the bill by calculating the exact cost of the fixed power charge for the $30$ days, performing the multiplications with all their decimals.
2. b) If the subtotal of the bill for that month (before applying the electricity tax) was exactly $60,27\text{ €}$, set up and solve the inverse operations needed to deduce how many $\text{kWh}$ the family consumed during that period.
3. c) Calculate the amount of the electricity tax, rounding the final result to the nearest euro cent (two decimal places). Then, justify why electricity supply companies set unit prices with three or more decimal places (thousandths of a euro, such as $0,148\text{ €}$) instead of rounding them from the start to cents ($0,15\text{ €}$).