Operations with decimal numbers
At a farmers' market in Galway, you mix $2.5$ kg of nuts at €14.20/kg with $1.75$ kg of dried fruit at €12.60/kg: what price per kilogram should you sell the mixture at to make a profit? You already know how to operate with decimals one at a time; now it is time to chain them together in real problems, without losing the order of operations.
Review: the four operations with decimals
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…
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.
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…
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.
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: a)…
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:
- a) $45.8 \times 100 =$
- b) $125.3 : 10 =$
- c) $0.75 \times 1000 =$
- d) $34.2 : 100 =$
Combined operations: the order of operations does not change with decimals
With decimals you work in the same order as with whole numbers, BIMDAS: Brackets, Indices (powers) and roots, Multiplication and Division (left to right), then Addition and Subtraction (left to right).
Apply the order of operations to solve the following expression with decimal numbers. Remember to carry out the operation inside brackets first: $$(4.5 + 2.1) \times 0.5…
Apply the order of operations to solve the following expression with decimal numbers. Remember to carry out the operation inside brackets first:
$$(4.5 + 2.1) \times 0.5 - 1.25$$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…
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$).
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$
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$
Multi-step money problems
Real-life problems chain together several operations: purchases, fees, rounding, splitting costs. Before calculating, write down the whole expression.
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…
Kenia has $154.20$ € in her bank account. During the week she carries out the following transactions:
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Pays an internet bill of $32.45$ €.
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Receives a mobile transfer from her grandmother of $20.50$ €.
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Buys three identical books for a total of $41.85$ €.
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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 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…
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?
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…
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:
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Subscription to a gaming channel: $12.45$ €.
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Afternoon snack with friends: $8.12$ €.
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Phone case: $5.67$ €.
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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$ €:
$$150 - (12.45 + 8.12 + 5.67 + 1.23 + 2.25) - H$$Where $H$ is the total sum of the roundings sent to the digital piggy bank.
Decimals in areas, mixtures and speeds
The same care with combined operations applies to formulae: areas involving $\pi$, prices of mixtures, or speed from distance and time.
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…
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?
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…
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 $09:39\text{ h}$.
- 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.
- b) Apply the formula $v = \frac{\text{distance}}{\text{time}}$ using the time in decimal hours to calculate the train's average speed in $\text{km/h}$.
- 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.
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…
A racing drone has travelled a distance of $14.58$ kilometres in $12$ minutes and $45$ seconds.
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Express the total flight time in hours using decimal numbers (round to four decimal places).
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Calculate the average speed of the drone in km/h using the previous value.
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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?
Back to the start: The total cost of the mixture is $2.5 \times 14.20 + 1.75 \times 12.60 = 35.50 + 22.05 = 57.55$, so the mixture costs €57.55 for $4.25$ kg. Dividing, it works out at $57.55 \div 4.25 \approx 13.54$, so about €13.54/kg. You must sell it for more than €13.54/kg to make a profit.