Exercises: Operations with decimal numbers

Cian is saving up to buy a video game. On Monday he had $24.50$ € in his piggy bank. On Tuesday his aunt gave him $15.75$ € and on Wednesday he decided to buy a phone case costing $12.30$ €. How much money does Cian have in his piggy bank right now? Perform the calculation lining up the decimal points correctly.

In Design and Technology class, Aoife is measuring several components for a robotics project. She has placed four identical pieces in a row, as shown in the diagram. If each piece measures $3.25$ cm long, what is the total length of the row?

A monthly subscription to a music streaming service costs $9.95$ €. If a group of 10 friends decides to create a family account that costs exactly 10 times that amount per year, what will be the total annual cost of the family subscription? Solve by moving the decimal point to the right.

In the school canteen, a large bag of mixed nuts costs $3.45$ €. Three friends decide to buy it to share and split the cost equally. How much money should each person contribute? Perform the division making sure you get two exact decimal places.

Calculate the following products and quotients by powers of 10 mentally. Remember that you only need to move the decimal point to the left or right as appropriate:

1. a) $45.8 \times 100 =$
2. b) $125.3 : 10 =$
3. c) $0.75 \times 1000 =$
4. d) $34.2 : 100 =$

Apply the order of operations to solve the following expression with decimal numbers. Remember to carry out the operation inside brackets first:

A can of fizzy drink has a capacity of $0.33$ litres. If a cardboard box holds 12 cans, how many litres of drink are there in total in a full box? Work out the multiplication manually.

Aoife wants to put a square rug in her room. If the area covered by the rug is $5.29$ m², what is the length of the side of the rug? Calculate the exact square root of the decimal number. (Hint: Think which integer multiplied by itself gives $529$).

For a laboratory practical in Science class, Seán needs to determine the mass of water evaporated from a beaker. The initial mass of the set was $12.5\text{ g}$ and, after heating it with the burner, the final mass is $3.48\text{ g}$. In his notebook, Seán writes:

1. a) Explain in detail what conceptual error Seán made regarding the place value of the decimal digits and the value of the hundredths.
2. b) Why is it necessary to add a zero to the minuend before operating?
3. c) Correct the calculation and find the actual mass of evaporated water.

In a local fruit shop you buy $1.85\text{ kg}$ of apples at $1.95$ €/kg and $0.48\text{ kg}$ of walnuts at $9.80$ €/kg. You have a $10$ € banknote in your pocket.

1. a) Before doing the precise calculation, make a mental estimate of the total amount by rounding the kilograms and unit prices to simple whole numbers (or halves). Based on your estimate, reason whether you will have enough money with the $10$ € note.
2. b) Set up a single combined expression and calculate the exact cost of the purchase.
3. c) Check if your estimate was accurate by finding the difference between the actual and estimated expenses, and calculate the exact change you should receive.

Calculate the result of the following expression according to the order of operations and the rule of signs:

$15.8 - 2.5 \times (4.2 - 3.75) + (1.24 + 0.36) : 0.4$

A streamer is analysing the data consumption of their recent live streams. In the first hour they used $2.45$ GB, in the second hour $1.8$ GB, and in the third, due to a technical glitch, only $0.95$ GB. If they have a monthly plan of $15$ GB and had already spent $6.25$ GB before starting these streams, how many gigabytes do they have left in their data plan?

Complete the gaps in the pyramid so that the value of each box is the sum of the two numbers directly below it.

An architect is designing a rectangular play area in a local park. The dimensions of the ground are $12.45$ metres long by $8.6$ metres wide. If she wants to enclose the entire perimeter with a protective fence costing $15.75$ € per linear metre, what will be the total budget for the fence?

In a swimming competition, the times of the top three swimmers in the $100$-metre freestyle event were:

• Swimmer A: $48.25$ seconds.

• Swimmer B: $49.12$ seconds.

• Swimmer C: $48.09$ seconds.

Calculate the time difference between the gold medallist and the bronze medallist. What is the mean time of the three swimmers? Round the mean to two decimal places.

A greengrocer buys $50$ kg of loose cherries for a total of $142.50$ €. When inspecting the goods, he discovers that $4.5$ kg are damaged and cannot be sold. If he wants to make a net profit of $60$ € on the total purchase, at what price per kilogram must he sell the cherries that are in good condition?

A water tank has a capacity of $120.5$ litres. A tap is turned on that pours in $4.25$ litres per minute. Simultaneously, the tank has a leak through which $0.75$ litres per minute are lost. If the tank initially contained $15.25$ litres, how many litres will it have after $12$ minutes of operation?

Kenia has $154.20$ € in her bank account. During the week she carries out the following transactions:

• Pays an internet bill of $32.45$ €.

• Receives a mobile transfer from her grandmother of $20.50$ €.

• Buys three identical books for a total of $41.85$ €.

• Withdraws $50$ € from an ATM to spend over the weekend.

If the bank charges her a maintenance fee of $2.15$ € at the end of the week, what is her final balance? Is her account overdrawn (negative balance)? Write the result using integers and decimals.

A group of $4$ friends goes to have an afternoon snack at a central café. Each one orders a hot chocolate for $2.35$ € and a scone for $1.65$ €. To anticipate the total bill:

• Aoife proposes adding the cost of an individual snack and multiplying the sum by $4$: $4 \times (2.35 + 1.65)$.

• Seán proposes calculating the total cost of the $4$ chocolates, that of the $4$ scones, and adding both amounts: $4 \times 2.35 + 4 \times 1.65$.

1. a) Carry out the calculations step by step using Aoife's method and Seán's method, checking whether both lead to the same result.
2. b) What mathematical property justifies that both methods yield exactly the same result?
3. c) Looking at the decimal digits of the prices, justify which of the two procedures is much more advantageous for doing quick mental math without making mistakes.

An Irish Rail InterCity train travels the $147\text{ km}$ separating two cities on a medium-distance route. The train departs at $08:15\text{ h}$ and arrives at its destination at $09:39\text{ h}$.

1. a) Calculate the total duration of the trip in minutes. Then, convert that time to hours in decimal form and explain in detail why it would be a serious error to divide the distance by $1.24$ to find the speed.
2. b) Apply the formula $v = \frac{\text{distance}}{\text{time}}$ using the time in decimal hours to calculate the train's average speed in $\text{km/h}$.
3. c) If the train consumes an average of $0.18$ litres of fuel per kilometre and the price per litre of fuel is $1.45$ €, write a single combined operation with decimal numbers to calculate the total fuel expenditure for the journey and solve it.

An automatic savings app rounds each of your purchases up to the nearest euro and saves the difference in a digital piggy bank. During an afternoon of shopping, you made the following payments:

• Subscription to a gaming channel: $12.45$ €.

• Afternoon snack with friends: $8.12$ €.

• Phone case: $5.67$ €.

• Sticker pack: $1.23$ €.

Calculate the value of the following expression to find how much money you have left in your bank account if you initially had $150$ € and, in addition to the purchases, the bank charged you a maintenance fee of $2.25$ €:

Where $H$ is the total sum of the roundings sent to the digital piggy bank.

The new logo for an e-sports team consists of an annulus (the shaded area between two concentric circles). The designer needs to know the exact area to order the vinyl stickers. The radius of the larger circle measures $6.25$ cm and the radius of the inner circle is $3.5$ cm.

Using as the value of $\pi \approx 3.1416$, calculate the area of the shaded region working with all decimals during the calculations, and round the final result to two decimal places.

A racing drone has travelled a distance of $14.58$ kilometres in $12$ minutes and $45$ seconds.

1. Express the total flight time in hours using decimal numbers (round to four decimal places).

2. Calculate the average speed of the drone in km/h using the previous value.

3. If the circuit record is a speed of $70.25$ km/h, by how many kilometres per hour did the drone fall short of matching the record?

In a bulk dried fruit and nut shop, they want to prepare a special "Energy" mix. They mix $2.5$ kg of walnuts at $14.20$ €/kg, $1.75$ kg of almonds at $12.60$ €/kg, and $0.75$ kg of raisins at $6.45$ €/kg.

Due to handling, it is estimated that the total weight of the mix is reduced by $0.05$ kg due to the loss of shells and dust. At what price per kilogram should they sell the resulting mix to obtain a total profit of $15.50$ € over the cost of the ingredients?

An architect designs a swimming pool in the shape of a cuboid whose internal dimensions are $5.25$ m long, $3.40$ m wide and $1.85$ m deep.

1. Calculate the maximum volume of water the pool can hold in cubic metres.

2. If the architect estimates the volume by truncating each dimension to one decimal place, what is the estimated volume?

3. Calculate the absolute error made between the actual volume and the volume estimated by truncation.

A hummingbird's heart beats approximately $1.2 \times 10^3$ times per minute at rest.

1. How many times does its heart beat in a full day? Express the result in standard form.

2. If the average life expectancy of this hummingbird is $5.5$ years, how many beats will it make in total throughout its life? (Assume a year has $365.25$ days).

3. If one beat displaces $0.025$ millilitres of blood, how many litres of blood does it pump in one hour?

Solve the following combined operation respecting the order of operations and working with generating fractions when recurring numbers appear. Express the final result in decimal form rounded to three decimal places:

On a domestic electricity bill in Ireland, the total amount payable before taxes was $48.60$ €. This amount is broken down into:

• A fixed power charge of $13.50$ €.

• A variable consumption charge depending on the kilowatt-hours ($\text{kWh}$) consumed.

During the billing month, the household consumed exactly $260\text{ kWh}$. We know that the unit price per $\text{kWh}$ is a decimal number with three decimal places of the form $0.1ab$ €/kWh, where $a$ and $b$ are two unknown digits.

1. a) State the equation or expression to calculate the unit cost per $\text{kWh}$, calculate its value, and determine which digits $a$ and $b$ represent.
2. b) The following month, the unit price per $\text{kWh}$ increases by $0.015$ €. If the family wants the total cost of the variable consumption (excluding the fixed charge) not to exceed $32$ €, what is the maximum whole number of $\text{kWh}$ they can consume? Justify your answer showing the appropriate division between decimals.

An athlete is training to run the San Silvestre Vallecana on a $10\text{ km}$ course, aiming to finish in a total time strictly under $45\text{ minutes}$. When passing the $5\text{ km}$ mark (halfway through the race), her stopwatch shows $23\text{ minutes and } 15\text{ seconds}$.

1. a) Convert the time of the first $5\text{ km}$ into exact decimal hours and calculate her average speed in that section in $\text{km/h}$.
2. b) What is the maximum time she has left to complete the last $5\text{ km}$ if she wants to achieve her goal of finishing under $45\text{ minutes}$? Express that time in exact decimal hours and calculate the minimum average speed required in the second half (rounded to two decimal places).
3. c) Her coach states: "To find the overall average speed of the entire race, it is enough to calculate the arithmetic mean of the two average speeds obtained." Numerically prove whether the coach's statement is correct or not, and explain why speeds cannot be directly averaged when the times spent on each section are different.