Exercises: Fractions. Equivalent fractions.

Look at the following figure and determine what fraction the shaded part represents. Then, write an equivalent fraction with a denominator of $16$.

Calculate the missing number so that the following pair of fractions is equivalent:

Check whether the following fractions are equivalent using the method of cross-multiplication:

Show all your working.

Cian has broken a chocolate bar into $12$ equal squares and has eaten $4$. His friend Aoife has an identical bar but broken into only $3$ big pieces. How many pieces should Aoife eat to have eaten the same amount of chocolate as Cian? Express it using equivalent fractions.

In the figure, the coloured squares are the ones Cian has eaten. Aoife's bar is uncoloured.

Obtain three fractions equivalent to $\frac{2}{7}$ by amplification. To do this, multiply the numerator and the denominator by $2$, by $3$, and by $10$.

Simplify the following fraction to its simplest form:

State which number you divided the numerator and denominator by at each step.

Look at the position of point $P$ on the number line. What fraction does it represent? Write an equivalent fraction to it with a denominator of $10$.

In a basketball team, $6$ out of the $15$ players are left-handed.

1. a) Write the fraction of left-handed players.
2. b) Which of the following fractions is equivalent to the one above?

• $\frac{1}{3}$

• $\frac{2}{5}$

• $\frac{3}{10}$

Cian is looking for an equivalent fraction to $\frac{2}{5}$ whose denominator is $15$. In his notebook, he has written the following: "To go from the denominator $5$ to $15$, I have to add $10$. Therefore, I also add $10$ to the numerator, so the equivalent fraction is $\frac{12}{15}$". Explain with reasoning why Cian's procedure is incorrect, relying on the concept of amplification. Then, correct his mistake, calculate the correct equivalent fraction, and formally prove that it is equivalent to $\frac{2}{5}$ using the cross-multiplication rule.

In a survey on sustainable mobility carried out at a secondary school, $18$ out of the $40$ students in a 1st Year group state that they walk or cycle to school.

1. a) Before doing any exact calculation, make a reasoned estimate of whether that proportion represents more than half, less than half, or exactly half of the group, using the fraction $\frac{1}{2}$ as a reference.
2. b) Simplify the fraction $\frac{18}{40}$ to its irreducible fraction and check whether your initial estimate was correct.
3. c) If there are $200$ students in the entire 1st Year and exactly the same proportion were maintained, how many students would walk or cycle? Verify that the new fraction obtained is equivalent to the irreducible one by applying cross-multiplication.

Two friends, Cian and Aoife, are comparing how much progress they have made in reading the same book. Cian says he has read $\frac{3}{4}$ of the total, while Aoife claims to have completed $\frac{9}{12}$ of the pages. Use visual representation to check whether both have read the same proportion of the book and justify your answer using cross-multiplication.

Calculate the value of the unknown $x$ so that the following pairs of fractions are equivalent:

1. a) $\dfrac{x}{15} = \dfrac{4}{5}$

1. b) $\dfrac{3}{7} = \dfrac{21}{x}$

1. c) $\dfrac{18}{x} = \dfrac{6}{11}$

In a video game, a character's energy bar is originally divided into $8$ equal segments. If the character currently has $\frac{6}{8}$ of their energy, how many segments would be full if we decided to divide the same bar into $24$ smaller parts? Write the corresponding equivalent fraction.

In the figure, the top bar is the original one (with $6$ of $8$ segments full) and the bottom one is the same bar divided into $24$ equal parts, still uncolored.

Find the irreducible fraction of the following expressions using successive division or by calculating the Highest Common Factor ($M.C.D.$):

1. a) $\dfrac{120}{180}$

1. b) $\dfrac{45}{105}$

1. c) $\dfrac{256}{512}$

From the following list of fractions, group together those that are equivalent to each other:

$\dfrac{1}{3}, \dfrac{5}{10}, \dfrac{2}{6}, \dfrac{1}{2}, \dfrac{4}{12}, \dfrac{3}{6}, \dfrac{10}{30}, \dfrac{7}{14}$

To make a fruit smoothie, a recipe states that $\frac{2}{5}$ of a litre of milk is needed. If you only have a measuring jug marked in millilitres ($1$ litre = $1000$ ml), which fraction with denominator $1000$ would be equivalent to $\frac{2}{5}$? Exactly how many millilitres of milk should you add?

Complete the following chains of equivalent fractions:

1. a) $\dfrac{2}{3} = \dfrac{4}{\square} = \dfrac{\square}{15} = \dfrac{20}{\square}$

1. b) $\dfrac{60}{48} = \dfrac{30}{\square} = \dfrac{\square}{12} = \dfrac{5}{\square}$

Indicate which of the following statements are true and which are false, explaining your answer in your notebook:

1. a) If we multiply the numerator of a fraction by $2$ and the denominator by $2$, we get an equivalent fraction.

1. b) When we add $3$ to the numerator and $3$ to the denominator of the fraction $\frac{1}{2}$, we get another equivalent fraction.

1. c) The fraction $\frac{17}{13}$ is irreducible because both the numerator and the denominator are prime numbers.

Based on data published by Met Éireann regarding spring in two Irish agricultural regions, it is known that in region A it rained on $36$ days out of a total of $90$ observed days, while in region B it rained on $24$ days during a recorded period of $60$ days.

1. a) State the fraction of rainy days with respect to the total number of days in each of the two regions.
2. b) Without calculating the associated decimal numbers, prove using the cross-multiplication rule whether the relative frequency of rainy days was exactly equivalent in both areas.
3. c) Calculate the irreducible fraction shared by both regions by dividing the numerator and denominator by their G.C.D., and explain what practical meaning the numerator and denominator of that irreducible fraction have for a farmer in the area.

Two students need to find out whether the fractions $\frac{72}{108}$ and $\frac{60}{90}$ represent exactly the same quantity, but they cannot agree on how to check it:

• Aoife proposes checking it directly using cross-multiplication ($72 \times 90$ and $108 \times 60$).

• Seán prefers simplifying each fraction by dividing its terms by the Greatest Common Divisor (G.C.D.) to compare them in their irreducible form.

1. a) Apply Aoife's method and determine whether they are equivalent.
2. b) Apply Seán's method by calculating the G.C.D. of each fraction and finding their irreducible fractions.
3. c) Compare both approaches: which of the two methods do you consider safer and more advantageous if you have to perform the calculations by hand and without a calculator? Justify your answer by analysing what happens to the size of the operations in each case.

In a graphic design studio, two image formats are being compared. The first format occupies $\frac{5}{8}$ of a card's memory, and the second occupies $\frac{7}{12}$. For a new project, you need to find an equivalent fraction to each of them that has the least common multiple of both denominators as its denominator. Once found, find a natural number $x$ such that the fraction $\frac{x}{48}$ is located exactly between the two original fractions.

Three friends, Aoife, Seán, and Niamh, bought a rectangular plot of land to grow an urban garden. Aoife gets $\frac{2}{5}$ of the land, Seán $\frac{4}{15}$, and Niamh the remainder.

1. a) Calculate the simplified fraction corresponding to Niamh's share.
2. b) Find an equivalent fraction to Niamh's share whose numerator is double the numerator of Seán's share.
3. c) If the total land measures $300$ square metres, is it true that Seán's share is equivalent to having $20$ plots of $\frac{4}{1}$ square metres each? Justify your answer.

Find the value of the unknown $n$ in each of the following equations so that the fractions are equivalent, knowing that $n$ must be a positive integer:

1. a) $\frac{n+2}{12} = \frac{15}{18}$

1. b) $\frac{8}{n-3} = \frac{32}{40}$

1. c) $\frac{3}{7} = \frac{n^2}{21}$

An architect divides a rectangular picture window into $8$ equal sections (arranged in two rows of four). Three of these sections are shaded and represent the part that will have tinted glass. Determine:

1. a) What irreducible fraction of the total does the shaded area represent?
2. b) Find a fraction equivalent to the previous one whose denominator is equal to the denominator of the irreducible fraction obtained in the previous section multiplied by $3$.

We have a mystery fraction. We know that it is equivalent to $\frac{45}{105}$. In addition, the following clues are given:

1. Its numerator is a two-digit number that is a multiple of $6$.

2. If we add its numerator and its denominator, the result is a number less than $100$.

Which fractions satisfy all the clues? Find all of them and show the steps taken.

In a robotics competition, Aoife's robot has completed $\frac{18}{24}$ of the track, while Seán's has completed $\frac{25}{30}$.

1. a) Simplify both fractions to their irreducible form to find out which robot is ahead.
2. b) Aoife claims that if her robot covers a further section equivalent to $\frac{1}{12}$ of the total, the fractions for both robots will become equivalent. Is her statement true? Prove it mathematically.

We want to construct a chain of equivalent fractions starting with $\frac{12}{16}$.

1. a) Write the irreducible fraction of the chain.
2. b) Find the missing term in the following equality: $\frac{12}{16} = \frac{x}{x+5}$. (Hint: use the property that cross-products of equivalent fractions are equal).
3. c) If we multiply the numerator of the irreducible fraction by a prime number $p$ and the denominator by the next prime number, could the new fraction obtained be equivalent to the original one? Justify your answer with an example.

A juice company uses two different containers. Container A is filled to $\frac{3}{4}$ of its capacity and container B to $\frac{12}{16}$ of its own.

1. a) Show graphically and by using cross products whether the filling fractions are equivalent.

1. b) If container A has a capacity of $2$ liters and container B has a capacity of $1.5$ liters, do they contain the same amount of juice in liters? Do they still represent the same proportion with respect to their own containers? Explain the difference between "equivalent fraction" and "absolute amount".

Find an unknown fraction from the following clues about its terms (both are natural numbers):

1. The fraction is not simplified; its denominator exceeds the numerator by $24$ units.

2. The sum of its numerator and its denominator is equal to $96$.

3. When simplifying it to its irreducible form, the sum of the new terms is equal to $8$.

1. a) Set up the relations above to deduce the value of the numerator and denominator of the original unsimplified fraction.
2. b) Obtain its irreducible fraction by dividing its terms by the G.C.D. and verify that it satisfies the third condition.
3. c) Determine the exact factor by which the irreducible fraction was amplified to generate the initial fraction, and justify, using reduction to a common denominator, why both fractions strictly represent the same part of the unit.

In a quality audit of Irish Rail's DART commuter rail service, delays on two lines are analysed over a quarter:

• On line C-1, $140$ trains were inspected, of which $119$ arrived on time at their destination.

• On line C-2, $260$ trains were inspected, of which $221$ arrived punctually.

1. a) Prove using the equality of cross-products whether both lines had an identical level of punctuality.
2. b) Reduce the punctuality proportion to its irreducible fraction using the G.C.D. and interpret what both numbers represent for someone who travels regularly on those trains.
3. c) Looking ahead to the next quarter, a third line (C-3) is planned to run a total of $450$ trains.

1. Mathematically justify why it is impossible for line C-3 to record exactly the same irreducible punctuality proportion as lines C-1 and C-2 if all trains must be considered whole units.

2. If $500$ trains were audited instead of $450$, what is the maximum number of delayed trains that could be allowed to match the punctuality proportion of the previous lines? Justify your reasoning through fraction amplification.