Equivalent Fractions

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

Show all your working.

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?

• $\frac{1}{3}$

• $\frac{2}{5}$

• $\frac{3}{10}$

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.

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 $10$.

Complete the following chains of equivalent fractions:

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

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

Simplify the following fraction to its simplest form:

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

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) $\dfrac{256}{512}$

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 $3$.

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

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

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.

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.