Exercises: Operations with fractions

At a snack with friends, several pizzas of the same size were bought, all cut into 8 equal slices.

• Marcos ate $\frac{3}{8}$ of a pizza.

• Sara ate $\frac{2}{8}$ of a pizza.

Calculate what fraction of a pizza they have eaten altogether and how many slices remain if only one pizza had been opened.

For an Art project, a group of students has a box with $60$ felt-tip pens. If they have used $\frac{2}{5}$ of the felt-tip pens to make a mural, how many felt-tip pens have they used in total?

Calculate the following additions and subtractions of fractions with the same denominator, simplifying the result where possible:

1. a) $\frac{5}{12} + \frac{1}{12}$

1. b) $\frac{11}{15} - \frac{4}{15}$

1. c) $\frac{7}{9} + \frac{2}{9} - \frac{4}{9}$

A young influencer spends her afternoon organizing her videos. She spends $\frac{1}{4}$ of the time recording, $\frac{1}{3}$ editing, and the rest of the time replying to comments. Calculate what fraction of the total time she spends recording and editing (remember to find a common denominator).

Perform the following fraction multiplications and simplify the results to the irreducible fraction:

1. a) $\frac{2}{3} \cdot \frac{5}{7}$

1. b) $\frac{3}{4} \cdot \frac{2}{5}$

1. c) $5 \cdot \frac{3}{10}$

In the kitchen of a café they want to share out $3$ litres of orange juice into glasses of $\frac{1}{4}$ of a litre each. How many glasses can they fill completely? Illustrate your reasoning by dividing each litre into parts. In the figure you have the $3$ litres in three $1$ L bottles; you can copy it into your notebook and divide each bottle.

Solve these divisions using cross-multiplication:

1. a) $\frac{4}{5} : \frac{2}{3}$

1. b) $\frac{1}{6} : \frac{1}{2}$

1. c) $\frac{3}{7} : 3$

A tablet battery indicates that it has $\frac{7}{10}$ of charge remaining. If after watching some videos it has consumed $\frac{2}{5}$ of the total charge, what fraction of battery does it still have left? (Hint: convert $\frac{2}{5}$ into a fraction with denominator $10$).

At a birthday party, two trays with slices of Spanish potato omelette of the same size are left over: on one tray there is $\frac{1}{2}$ of an omelette left and on the other there is $\frac{1}{4}$. Sergio wants to calculate how much omelette is left in total and writes in his notebook:

1. a) Without doing any written calculations, explain why Sergio's result is impossible in reality (think about whether, when combining food, it makes sense to end up with less than the half omelette you already had on the first tray).
2. b) Explain what Sergio's mathematical mistake was when working with the fractions.
3. c) Solve the addition correctly by finding a common denominator and express the result as an irreducible fraction.

To prepare an isotonic drink in the school lab, you pour into a $2$-litre jug first $\frac{4}{5}$ of a litre of mineral water and, then, you add $\frac{7}{10}$ of a litre of orange juice.

1. a) Estimate: Before calculating exactly, reason whether the jug will overflow (that is, whether the sum will exceed $2$ litres) and whether the final volume will be greater than, smaller than or equal to $1$ and a half litres ($\frac{3}{2}$ of a litre). Justify your estimate by mentally comparing each fraction with one whole or with one half.
2. b) Calculate: Find the exact amount of liquid in the jug by adding the fractions using the least common multiple. Express the result as an irreducible fraction.
3. c) Check: Convert your fraction into a decimal number and check whether it confirms your estimate from part a).

To make a tropical drink, a group of friends mixes $\frac{3}{4}$ of a litre of pineapple juice, $\frac{1}{3}$ of a litre of orange juice, and $\frac{5}{6}$ of a litre of sparkling water. If after mixing it all together, they drink $\frac{2}{5}$ of a litre before putting it in the fridge, how much drink is left in total? Express the simplified answer in litres.

A streaming platform divides its catalogue of series into three parts: $\frac{2}{5}$ are action, $\frac{1}{3}$ are comedy and the rest are documentaries. If a user has already watched $\frac{3}{4}$ of all the available action series, what fraction of the total catalogue is represented by the action series that they have not yet watched?

Calculate the result of the following mixed operation, respecting the order of operations and simplifying the final result to obtain the irreducible fraction:

A gardener has spent $\frac{2}{9}$ of the savings they had for tools on buying a lawnmower and $\frac{3}{10}$ of the initial budget on pruning shears. If they still have $172$ € left in their budget, how much money did they have at the start?

(Hint: First calculate what fraction of the total has been spent by adding both amounts).

Apply the laws of indices and roots to simplify the following numerical expression:

In the technology workshop, wooden battens are being cut. We have a batten $\frac{15}{2}$ metres long and we want to divide it into equal pieces measuring $\frac{3}{4}$ of a metre each. How many full pieces will we get? Will there be any wood left over?

Find the value of the unknown $x$ in the following equation with fractions:

A drone takes off from the ground and ascends to a height of $12$ metres. Then, due to a gust of wind, it descends $\frac{14}{3}$ of a metre. If it then rises again by a height equal to $\frac{2}{5}$ of the position it was in after the descent, at what height above the ground is the drone finally located?

In Maths class, the following combined operation is given to solve:

Two classmates defend different strategies:

• Sofía first multiplies the numerators and denominators inside the parentheses straight across, and then carries out the division by cross-multiplying the large numbers to simplify only at the very end.

• Lucía changes the division into a multiplication by the reciprocal, factors the numbers into prime factors to cancel common factors on the top and bottom before multiplying, and calculates at the end.

1. a) Solve the operation following Sofía's method step by step.
2. b) Solve the operation following Lucía's method, clearly showing how you simplify before operating.
3. c) Compare both paths: which of the two is more efficient and reduces the risk of making calculation errors with large numbers? Justify your answer.

We want to solve a logic puzzle with fractions: find two different unit fractions (that is, of the form $\frac{1}{a}$ and $\frac{1}{b}$, with numerators equal to $1$ and different natural denominators) that satisfy both conditions at the same time:

1. Their product is exactly $\frac{1}{12}$.

2. Their difference (the larger minus the smaller) is exactly $\frac{1}{12}$.

1. a) Reason out which numbers $a$ and $b$ must be by analysing the factors of the common denominator.
2. b) Perform the operations and verify that both fractions simultaneously satisfy the product and subtraction conditions.
3. c) Reason why it would be impossible for the sum of two positive fractions to yield a fraction smaller than either of the two initial fractions.

The publisher of a youth magazine is analysing where its readers come from. It knows that $\frac{2}{5}$ of the total live in Spain. Of the remaining readers, $\frac{3}{8}$ live in Latin American countries. If there are still $1.500$ readers who live in the rest of the world, how many readers does the magazine have in total?

Calculate the result of the following mixed operation, simplifying the answer to its irreducible fraction:

In a chemistry laboratory, container $A$ contains $\frac{3}{4}$ of a litre of alcohol, and container $B$ contains $\frac{2}{3}$ of a litre of water. Half of the contents of container $A$ is transferred into container $B$. What fraction of a litre of mixture is now in container $B$? If $\frac{1}{5}$ of the total mixture in $B$ is then spilled, how many litres remain in that container?

An architect is designing a sustainable house. The garden occupies $\frac{2}{7}$ of the total plot. The house occupies $\frac{3}{5}$ of the remaining area. If the garage and the swimming pool occupy the final $120$ $m^2$, what is the total area of the plot?

Find the value of the following expression involving powers and roots of rational numbers. Remember to apply the order of operations and simplify fully:

Three friends, Alba, Bruno and Carla, buy a giant pizza for dinner. Alba eats $\frac{1}{4}$ of the pizza. Bruno eats $\frac{1}{3}$ of what is left. Carla eats $\frac{2}{5}$ of what is left after Alba and Bruno have eaten.

1. a) What fraction of the pizza is left over?

1. b) If the pizza weighed $1,2$ kg, how many grams did each person eat?

A cyclist rides a stage of $120$ km in three legs. In the first hour, they cover $\frac{2}{5}$ of the journey. In the second hour, due to fatigue, their speed decreases and they only cover $\frac{1}{3}$ of the remaining distance. How many kilometres do they have left to cover for the third hour? What fraction of the total does that final leg represent?

Calculate the value of the unknown $x$ in the following equation with fraction operations:

According to management data from a river basin authority in Spain, a reservoir starts autumn with a certain reserve of stored water. During the month of October, the following changes are recorded:

• $\frac{3}{8}$ of the initial water is released for agricultural irrigation and the river's ecological flow.

• Of the water that remains in the reservoir after irrigation, $\frac{1}{5}$ evaporates due to late-season high temperatures.

• At the end of the month, a rainstorm adds a volume of water equivalent to $\frac{1}{4}$ of the water that was initially present at the beginning of the month.

After these changes, the reservoir holds exactly $150\text{ hm}^3$ (cubic hectometres) of water.

1. a) Write a single combined expression with fractions that represents the fraction of the initial water remaining just before the rainstorm began.
2. b) Determine what fraction of the initial volume is in the reservoir at the end of October.
3. c) Calculate how many cubic hectometres of water the reservoir contained at the beginning of the month.
4. d) Justify whether the reservoir has gained or lost water throughout the month compared to its initial level by comparing the resulting fraction to one whole.

A Renfe high-speed train departs from Madrid heading to Seville with all its seats occupied and makes only two intermediate stops along the way:

• At the first stop (Ciudad Real), exactly $\frac{1}{4}$ of the passengers get off and none board.

• At the second stop (Córdoba), an unknown fraction $\frac{a}{b}$ of the people travelling on the train at that moment get off, and again no one boards.

• The train arrives at its final destination in Seville with $120$ passengers, which represents exactly $\frac{2}{5}$ of the total number of passengers who began the trip in Madrid.

1. a) Deduce how many passengers in total were on the train when leaving Madrid.
2. b) Calculate how many people were travelling on the train right after departing from the Ciudad Real stop.
3. c) What fraction of the initial passengers from Madrid did the passengers who got off in Córdoba represent?
4. d) Determine the value of the unknown fraction $\frac{a}{b}$; that is, what fraction of the passengers on board the train at that moment got off in Córdoba. Express the result as an irreducible fraction and justify each step.

Topic challenge. How does your household manage its resources? Become an environmental auditor by analysing your home's water consumption and comparing it with official data.

1. Data collection: Check the INE website for the average water consumption in Spain (approx. $133$ litres per person per day) or record the water meter readings at your home on two consecutive days to calculate your actual daily usage.

2. Fractions and rationals: Estimate and categorise how water is distributed across activities (shower, toilet flush, cooking, cleaning). Express each usage as a fraction of the total and simplify it to its irreducible fraction, justifying why it belongs to the set of rational numbers ($\mathbb{Q}$). Also find two equivalent fractions for each case by expanding the values.

3. Calculation with decimals: In a spreadsheet, knowing that the average price of a $m^3$ ($1000$ litres) is $1,97$ €, use decimals to calculate the daily cost of each activity and the estimated bill for one month ($30$ days).

4. Reasoned proposal: Write a brief report proposing a plan to reduce consumption by $2/15$, and calculate the monthly savings in euros using decimals.

Assessment criteria:

• Rigorous derivation of irreducible and equivalent fractions from real data.

• Correctness in combined operations with decimal numbers.

• Clear organisation of data in the spreadsheet and feasibility of the proposal.