Fractions. Equivalent Fractions

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

Show all your working.

Two friends, Marcos and Lucía, are comparing how much progress they have made in reading the same book. Marcos says he has read $\frac{3}{4}$ of the total, while Lucía 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.

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".

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

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 $10$.

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.

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

Simplify the following fraction to its simplest form:

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

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?

• $\frac{1}{3}$

• $\frac{2}{5}$

• $\frac{3}{10}$

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:

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

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

1. a) Simplify both fractions to their irreducible form to find out which robot is ahead.
2. b) Irene 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.

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 $10$.

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

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

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

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) $\dfrac{18}{x} = \dfrac{6}{11}$

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.

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, Lucía, Marcos, and Sofía, bought a rectangular plot of land to grow an urban garden. Lucía gets $\frac{2}{5}$ of the land, Marcos $\frac{4}{15}$, and Sofía the remainder.

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

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.