Equivalent fractions

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

Show all your working.

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.

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".

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.

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 $10$.

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

Simplify the following fraction to its simplest form:

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

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.

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) $\dfrac{256}{512}$

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.

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.

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

Complete the following chains of equivalent fractions:

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

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

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