Exercises: Fractions. Equivalent fractions.

In a chocolate bar like the one in the picture, we have eaten the pieces shaded in orange.

1. a) Write down the fraction representing the pieces eaten relative to the whole bar.
2. b) Simplify the fraction above to its simplest form.

A group of friends is comparing the success of their social media posts. Aoife says that $2$ out of every $5$ followers liked her latest photo.

1. a) Write down the fraction of "likes" for Aoife.
2. b) Find an equivalent fraction by multiplying both terms by $3$.
3. c) Find another equivalent fraction if Aoife had a total of $50$ followers.

In a video game, you collected $45$ coins out of a total of $60$ available in the level. On the other hand, your brother collected $15$ out of a total of $20$ in his game.

Use the cross-products rule to check whether the two fractions of coins collected are equivalent.

Look at the following rectangles of equal size. The first is divided into $4$ equal parts and the second into $8$ equal parts.

1. a) Write down the shaded fraction of Figure A.
2. b) Write down the shaded fraction of Figure B.
3. c) Are they equivalent? Justify your answer by looking at the diagram and carrying out the calculation.

To make a fruit smoothie, the recipe states that we should use $\frac{3}{4}$ of a litre of milk. If we want to make a larger quantity while maintaining the ratio, and the new measuring jug marks eighths of a litre ($8$ parts):

Find the value of $x$ so that the fractions are equivalent:

In the school library, the librarian tells us that $\frac{12}{30}$ of the books are adventure books. Help him simplify this fraction to put on a poster:

1. a) Divide the numerator and the denominator by $2$.
2. b) Divide the previous result by $3$.
3. c) Is the fraction obtained in part b in its simplest form? Why?

Copy into your notebook and complete the missing value in each case so that the pairs of fractions are equivalent:

1. a) $\frac{1}{3} = \frac{\Box}{12}$

1. b) $\frac{10}{15} = \frac{2}{\Box}$

1. c) $\frac{\Box}{7} = \frac{20}{28}$

Identify, from the following list, which fractions are equivalent to $\frac{2}{3}$:

In a Mathematics exam, Aoife has to find a fraction equivalent to $\frac{4}{9}$ whose numerator is $12$. Aoife writes on her paper:

"Since to go from $4$ to $12$ I have to add $8$, I do the same in the denominator: $\frac{4}{9} = \frac{4+8}{9+8} = \frac{12}{17}$. Therefore, $\frac{12}{17}$ is equivalent to $\frac{4}{9}$".

1. a) Explain in detail why Aoife's procedure is incorrect, relying on the definition of fraction amplification and checking the cross-product of both fractions.
2. b) Correct her exercise by calculating the true fraction equivalent to $\frac{4}{9}$ with numerator $12$.

Look at the fractions $\frac{12}{24}$ and $\frac{21}{40}$.

1. a) Estimate: Without performing cross-products or written simplifications, mentally compare each of them with the benchmark value $\frac{1}{2}$ and deduce whether they can be equivalent or not. Explain your initial reasoning in your own words.
2. b) Check: Carry out the rigorous mathematical check using the cross-product rule to confirm whether your estimation was correct.

Two graphic designers are working with grids to create a logo. The first designer has shaded $\frac{12}{20}$ of their grid, while the second has shaded $\frac{18}{30}$ of theirs. Look at the following graphic representations and explain whether both have shaded the same proportion of area, by checking whether the fractions are equivalent.

In an esports competition, one player has completed $42$ of the $60$ levels of a video game. Another player has completed the same proportion of levels, but the game they are playing has a total of $90$ levels. How many levels has the second player completed? Solve this by finding the unknown term $x$ in the equivalent fractions equation.

Simplify the following fractions to their simplest form. To do this, use the method of dividing the numerator and denominator by their highest common factor (HCF):

1. a) $\dfrac{126}{180}$

1. b) $\dfrac{210}{315}$

1. c) $\dfrac{154}{198}$

An influencer analyses the impact of their videos. In the first one, they got $36$ "likes" for every $80$ views. In the second one, they got $45$ "likes" for every $100$ views. Use the cross-multiplication rule to determine if the success rate is equivalent in both videos.

Find the value of the unknown $n$ in each of the following equalities so that the fractions are equivalent:

1. a) $\dfrac{n}{15} = \dfrac{12}{45}$

1. b) $\dfrac{7}{n} = \dfrac{28}{36}$

1. c) $\dfrac{10}{18} = \dfrac{25}{n}$

From the following fractions, identify which one does not belong to the group of equivalent fractions. Justify your answer by simplifying all of them to their simplest form.

To carry out a chemistry experiment, a laboratory needs to mix liquids in a ratio of $\frac{3}{8}$. If they have a container with a total capacity of $120$ ml for the mixture, how much active liquid should they pour in if they want to maintain the same ratio? Represent the situation using two equivalent fractions where the denominator of the second is $120$.

Draw a number line in your notebook (mark 0 and 1, and use a unit that is big enough, for example 8 small squares) and represent the following fractions on it. Show graphically that they are equivalent by observing that they are all located at the same point on the line:

You can do it on paper and send a photo with your answer.

On Dublin commuter Line 5, Irish Rail conducts a punctuality audit during one week:

• On the section Maynooth - Connolly, $72$ of the $90$ scheduled trains arrived on time.

• On the section Connolly - Bray, $150$ trains were scheduled in total, and the company claims that the punctuality rate was exactly equivalent to that of the first section.

1. a) Model the punctuality proportion of the first section using a fraction and simplify it step by step until obtaining its irreducible fraction.
2. b) Set up an equality of equivalent fractions to find out how many trains had to arrive on time in the second section for the company's claim to be true. Use cross-products to solve it.

To check whether the fractions $\frac{84}{126}$ and $\frac{140}{210}$ are equivalent, two 2nd Year students propose different approaches:

• Seán proposes directly using the cross-product criterion by calculating $84 \times 210$ and $126 \times 140$.

• Aoife proposes simplifying both fractions beforehand to their irreducible fraction by dividing numerator and denominator by their greatest common divisor ($\text{gcd}$).

1. a) Analyze the advantages and disadvantages of each method taking into account the size of the numbers and the risk of making calculation errors without a calculator. Justify which of the two methods is more convenient in this case.
2. b) Apply Aoife's method to check whether the fractions are equivalent, indicating the $\text{gcd}$ of the terms of each fraction and the irreducible fraction obtained in each case.

In a robotics competition, two robots must cover the same distance, from the Start to the Finish. The first robot has completed $\frac{42}{105}$ of the course. The second robot has covered a fraction equivalent to the previous one, but its numerator and denominator are natural numbers whose sum is exactly $112$.

What fraction of the course has the second robot covered? Justify whether both robots are at the same point on the track and calculate the irreducible fraction that represents the position of both.

Find a fraction equivalent to $\frac{126}{198}$ that satisfies the following conditions simultaneously:

1. Its denominator is a number between $50$ and $100$.

2. The sum of the digits of the numerator and the sum of the digits of the denominator are each, separately, a multiple of $9$.

3. It is a proper fraction.

Explain step by step the process of simplifying and expanding (multiplying) you followed to find it.

An interface designer is working on the resolution of a new screen. The aspect ratio is $16:9$ (which means that for every $16$ units of width there are $9$ of height). Draw the diagram in your notebook. If the width of the screen is defined by a number $x$ and the height by a number $y$, find the values of $x$ and $y$ knowing that:

• $\frac{x}{y}$ is equivalent to $\frac{16}{9}$.

• The area of the rectangle formed ($x \times y$) is $576$ square units.

• $x$ and $y$ must be natural numbers.

You can do it on paper and send a photo with your answer.

Given the fraction $\frac{A}{B}$, we know it is equivalent to $\frac{4}{7}$. If we add $12$ units to the numerator, for the new fraction to be equivalent to the original one, the denominator must increase by $21$ units.

Prove whether this statement is true for any initial value of $A$ and $B$ that maintains the ratio $\frac{4}{7}$. Next, find a fraction equivalent to $\frac{4}{7}$ such that the product of its terms ($A \times B$) is $700$.

In a strategy video game, a character's health is represented by a bar. Currently, the character has $\frac{15}{24}$ of their total health. The player consumes a potion that increases current health by an amount such that the new fraction of health is equivalent to $\frac{11}{12}$ of total health.

If the denominator of the total health (maximum capacity) is $120$ points, how many health points exactly did the character recover with the potion? Express all fractions in the problem with a denominator of $120$ before solving it.

Three siblings share the cash prize of a mathematics competition as follows: the oldest receives $\frac{24}{60}$ of the total, the middle one receives a fraction equivalent to $\frac{18}{45}$, and the youngest gets the remainder.

1. Calculate the simplest form fraction of what each of the two older siblings receives.

2. Who receives more money, the oldest or the middle sibling?

3. What fraction of the total does the youngest sibling receive? Is it greater or less than their siblings' shares?

Find the value of the unknown $n$ so that the following fractions are equivalent:

Once you have found $n$, find another fraction equivalent to the previous ones whose denominator is the square of $n$. Is it possible for the numerator of this last fraction to be a prime number? Justify your answer.

Consider the purely recurring decimal $1,\widehat{36}$ and the mixed recurring decimal $1.3\widehat{6}$.

1. Find the generating fractions of both numbers and simplify them until they are irreducible.

2. Is either of these fractions equivalent to $\frac{60}{44}$?

3. Is there a natural number by which you must multiply the denominator of the generating fraction of $1,\widehat{36}$ to obtain the denominator of the generating fraction of $1.3\widehat{6}$? Justify your answer.

Determine two positive fractions $\frac{a}{b}$ and $\frac{c}{d}$ knowing that both have $\frac{5}{8}$ as their common irreducible fraction and that they satisfy the following independent requirements:

1. a) In the fraction $\frac{a}{b}$, the product of its numerator and its denominator is exactly $1000$. Deduce the natural scale factor by which the irreducible fraction has been amplified and write the fraction $\frac{a}{b}$.
2. b) In the fraction $\frac{c}{d}$, the sum of its numerator and its denominator is equal to the smallest common multiple of $13$ and $3$ that lies strictly between $100$ and $200$. Determine the corresponding amplification factor and write the fraction $\frac{c}{d}$.
3. c) Verify that the obtained fractions $\frac{a}{b}$ and $\frac{c}{d}$ are equivalent to each other by applying the cross-product rule.

The Department of the Environment, Climate and Communications publishes the following data on two reservoirs in the same river basin:

• The Poulaphouca reservoir has a maximum capacity of $160\text{ hm}^3$ and currently contains $64\text{ hm}^3$ of water.

• The Leixlip reservoir has an unknown total capacity $C$, but it is known that if it received an additional $15\text{ hm}^3$ it would reach a stored volume of $55\text{ hm}^3$. Furthermore, technicians confirm that the current proportion of water relative to its total capacity is exactly the same as in Poulaphouca (their fill fractions are equivalent).

1. a) Model the situation by setting up the equality of fractions between both reservoirs and calculate with reasoning the total capacity $C$ of Leixlip using cross-products.
2. b) During the following month, rainfall adds exactly $20\text{ hm}^3$ of water to each of the two reservoirs. A spokesperson claims: "Since exactly the same volume of water has entered both reservoirs, their fill proportions relative to their total capacities remain equivalent". Prove mathematically whether the spokesperson's claim is true or false by calculating the irreducible fraction of the new fill level of each reservoir.