Fractions. Equivalent fractions

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.

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.

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.

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 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?

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 $12$.

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) $\dfrac{154}{198}$

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.

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.

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:

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) $\frac{\Box}{7} = \frac{20}{28}$

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.

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) $\dfrac{10}{18} = \dfrac{25}{n}$

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 $120$.

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.

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.

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

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.

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.

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.