Equivalent Fractions and Simplest Form

In a chocolate bar like the one in the image, we have consumed the pieces that are colored orange.

1. a) Write the fraction that the consumed pieces represent with respect to the whole bar.
2. b) Simplify the previous fraction until you reach its irreducible fraction.

Look at the following rectangles of equal size. The first one is divided into $4$ equal parts and the second one 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 observing the drawing and performing the calculation.

Draw a number line in your notebook and represent the following fractions on it. Demonstrate graphically that they are equivalent by observing that they are all located at the same point on the line:

A group of friends is comparing the success of their social media posts. Ashley says that $2$ out of every $5$ followers have "liked" her latest photo.

1. a) Write Ashley's fraction of "likes".
2. b) Obtain an equivalent fraction by expansion by multiplying both terms by $3$.
3. c) Obtain another equivalent fraction if Ashley had a total of $50$ followers.

To prepare a fruit smoothie, the recipe states that we must use $\frac{3}{4}$ of a gallon of milk. If we want to prepare a larger quantity keeping the proportion and the new measuring jug marks eighths of a gallon ($8$ parts):

Find the value of $x$ so that the fractions are equivalent:

Given the fraction $\frac{A}{B}$, we know that it is equivalent to $\frac{4}{7}$. If we add $12$ units to the numerator, for the new fraction to be equivalent to the original one, the denominator must increase by $21$ units.

Prove whether this statement is true for any initial value of $A$ and $B$ that maintains the proportion $\frac{4}{7}$. Next, find a fraction equivalent to $\frac{4}{7}$ such that the product of its terms ($A \cdot B$) is $700$.

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 it 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 an irreducible fraction? Why?

Simplify the following fractions until you obtain their irreducible fraction. To do this, use the method of dividing the numerator and denominator by their greatest common factor (GCF):

1. a) $\dfrac{126}{180}$

1. b) $\dfrac{210}{315}$

1. c) $\dfrac{154}{198}$

Find a fraction equivalent to $\frac{126}{198}$ that meets 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 simplification and expansion you have followed to find it.

In a video game, you have collected $45$ coins out of a total of $60$ that were in the level. On the other hand, your brother has collected $15$ out of a total of $20$ in his game.

Use the cross-multiplication rule to check if the two fractions of collected coins are equivalent.

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

An influencer analyzes the impact of their videos. In the first one, they obtained $36$ "likes" for every $80$ views. In the second one, they obtained $45$ "likes" for every $100$ views. Use the cross-multiplication rule to determine if the success proportion is equivalent in both videos.

Copy into your notebook and complete the missing term 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, a 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 passed? Solve it by finding the unknown term $x$ in the equality of equivalent fractions.

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

In a robotics competition, two robots must cover the same distance. The first robot has completed $\frac{42}{105}$ of the route, while 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 route 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.

An interface designer is working on the resolution of a new screen. The aspect ratio is $16:9$ (which means that for every $16$ units of width there are $9$ of height). Draw the diagram in your notebook. If the width of the screen is defined by a number $x$ and the height by a number $y$, find the values of $x$ and $y$ knowing that:

• $\frac{x}{y}$ is equivalent to $\frac{16}{9}$.

• The area of the rectangle formed ($x \cdot y$) is $576$ square units.

• $x$ and $y$ must be natural numbers.

Find the value of the unknown $n$ so that the following fractions are equivalent:

Once $n$ is found, find another fraction equivalent to the previous ones whose denominator is the square of $n$. Is it possible for the numerator of this last fraction to be a prime number? Reason your answer.

Of 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 irreducible fraction.

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 the current health by an amount such that the new health fraction is equivalent to $\frac{11}{12}$ of the total health.

If the denominator of the total health (the maximum capacity) is $120$ points, exactly how many health points has the character recovered with the potion? Express all the fractions in the problem with a denominator of $120$ before solving it.

Three siblings share an inheritance in the following way: the eldest receives $\frac{24}{60}$ of the total, the middle one receives a fraction equivalent to $\frac{18}{45}$ and the youngest the rest.

1. Calculate the irreducible fraction of what each of the two older siblings receives.

2. Who receives more money, the eldest or the middle sibling?

3. What fraction of the total does the youngest sibling receive? Is it greater or less than their siblings' shares?

Consider the pure repeating decimal $1,\widehat{36}$ and the mixed repeating decimal $1,3\widehat{6}$.

1. Find the generating fractions of both numbers and simplify them until they are irreducible.

2. Are any of these fractions equivalent to $\frac{60}{44}$?

3. Find 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 $1,3\widehat{6}$ as a result.