Exercises: Fractions. Equivalent fractions.

Look at the following figure and determine what fraction the shaded part represents. Then, write an equivalent fraction that has the number $16$ as its denominator.

Find the value of the missing number so that the following pair of fractions are equivalent:

Check if the following fractions are equivalent using the cross products method:

Show the calculations performed.

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

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

Simplify the following fraction until you obtain its irreducible fraction:

Indicate what number you have divided the numerator and denominator by in each step.

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

In a basketball team, $6$ 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 previous one?

• $\frac{1}{3}$

• $\frac{2}{5}$

• $\frac{3}{10}$

Two friends, Marcus and Lucy, are comparing how far they have progressed in reading the same book. Marcus says he has read $\frac{3}{4}$ of the total, while Lucy claims to have completed $\frac{9}{12}$ of the pages. Use graphical representation to check if both have read the same proportion of the book and justify your answer using cross-multiplication.

Find 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.

Find the irreducible fraction of the following expressions using the successive division method or by calculating the Greatest 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 those that are equivalent to each other:

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

Hint: Simplify each fraction to its irreducible form to find them more EASILY.

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

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, justifying your answer in your notebook:

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

1. b) By adding $3$ to the numerator and $3$ to the denominator of the fraction $\frac{1}{2}$, we obtain 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 takes up $\frac{5}{8}$ of a card's memory, and the second takes up $\frac{7}{12}$. For a new project, it is necessary to find an equivalent fraction to one of them that has the least common multiple of both denominators as its denominator. Once found, look for a natural number $x$ such that the fraction $\frac{x}{24}$ is located exactly between the two original fractions.

Three friends, Lucy, Marcus, and Sophia, have bought a rectangular plot to cultivate a community garden. Lucy keeps $\frac{2}{5}$ of the land, Marcus $\frac{4}{15}$ and Sophia the rest.

1. a) Find the simplified fraction that corresponds to Sophia.
2. b) Find a fraction equivalent to Sophia's part whose numerator is double the numerator of Marcus's part.
3. c) If the total land measures $300$ square feet, is it true that Marcus's part is equivalent to having $20$ pieces of $\frac{4}{1}$ square feet each? Justify your answer.

Find the value of the unknown $n$ in each of the following equalities 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) $\frac{3}{7} = \frac{n^2}{147}$

An architect divides a rectangular window into several sections. The shaded area represents the part that will have tinted glass. Look at the following diagram and 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 sum of the numerator and denominator of the irreducible fraction obtained in the previous part multiplied by $3$.

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

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, Irene's robot has completed $\frac{18}{24}$ of the circuit, while Alexander's has completed $\frac{25}{30}$.

1. a) Simplify both fractions to their irreducible fraction to find out which robot is in the lead.
2. b) Irene claims that if her robot covers one more section equivalent to $\frac{1}{12}$ of the total, the fractions of both robots will become equivalent. Is her claim true? Prove it mathematically.

We want to build 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 in equivalent fractions the cross products 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 capacity.

1. a) Prove graphically and using cross products whether the filling fractions are equivalent.
2. b) If container A has a capacity of $2$ gallons and container B has a capacity of $1,5$ gallons, do they contain the same amount of juice in gallons? Do they still represent the same proportion with respect to their own containers? Explain the difference between "equivalent fraction" and "absolute amount".