Fractions and Equivalent Fractions
Marcus says he has read $\frac{3}{4}$ of his book. Lucy says she has read $\frac{9}{12}$ of hers. The books are the same size. Who is farther along? The numbers look different, but the answer may surprise you: these are equivalent fractions.
What Are Equivalent Fractions?
Reading a fraction
A fraction tells you how many equal parts you have out of how many equal parts make the whole. The bottom number (denominator) says how many equal parts the whole is cut into. The top number (numerator) says how many of those parts you are talking about.
The same idea works for a group: if 6 of the 15 players on a basketball team are left-handed, then $\frac{6}{15}$ of the team is left-handed. It also works on a number line: cut the space between $0$ and $1$ into equal steps, and count how many steps the point is from $0$. The parts must always be equal.
Same amount, different names
Half a pizza, two-fourths, or four-eighths: it is the same amount of pizza, even though it is cut differently.
How can you tell without drawing? Use cross products.
Check if the following fractions are equivalent using the cross products method: $$\frac{4}{6} \text{ and } \frac{10}{15}$$ Show the calculations performed.
Check if the following fractions are equivalent using the cross products method:
$$\frac{4}{6} \text{ and } \frac{10}{15}$$Show the calculations performed.
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…
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.
In a basketball team, $6$ of the $15$ players are left-handed. a) Write the fraction of left-handed players. b) Which of the following fractions is equivalent to the…
In a basketball team, $6$ of the $15$ players are left-handed.
- a) Write the fraction of left-handed players.
- b) Which of the following fractions is equivalent to the previous one?
-
$\frac{1}{3}$
-
$\frac{2}{5}$
-
$\frac{3}{10}$
Making Equivalent Fractions by Multiplying
If you multiply the numerator and the denominator by the same number, you get an equivalent fraction. You cut every piece into smaller pieces, so you have more pieces, but each one is smaller.
This is also how you change the size of the pieces. Suppose a video game energy bar has $8$ segments and $6$ are full, so you have $\frac{6}{8}$. To cut the bar into $24$ segments, notice $8 \cdot 3 = 24$. Multiply the top by $3$ as well: $\frac{6}{8} = \frac{18}{24}$. The bar is just as full.
All equivalent fractions of $\frac{3}{7}$ look like $\frac{3k}{7k}$ for some whole number $k$: $\frac{3}{7}, \frac{6}{14}, \frac{9}{21}, \frac{12}{28}, \dots$ That pattern is handy when a problem gives you clues about a "mystery fraction."
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 $10$.
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 $10$.
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 $10$.
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 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…
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.
Simplifying to Lowest Terms
Going the other way, you can divide the numerator and the denominator by the same number. This is called simplifying. When you cannot go any further, the fraction is in simplest form (also called lowest terms).
You do not have to find the GCF right away. You can also divide step by step by any common factor you spot (like $2$, $3$, or $5$) until nothing common is left.
Simplifying is also the quickest way to check whether fractions are equivalent: simplify each one, and equivalent fractions end up identical. For example, $\frac{4}{12}$ and $\frac{10}{30}$ both simplify to $\frac{1}{3}$.
Simplify the following fraction until you obtain its irreducible fraction: $$\frac{18}{24}$$ Indicate what number you have divided the numerator and denominator by in…
Simplify the following fraction until you obtain its irreducible fraction:
$$\frac{18}{24}$$Indicate what number you have divided the numerator and denominator by in each step.
Find the irreducible fraction of the following expressions using the successive division method or by calculating the Greatest Common Factor ($M.C.D.$): a)…
Find the irreducible fraction of the following expressions using the successive division method or by calculating the Greatest Common Factor ($M.C.D.$):
- a) $\dfrac{120}{180}$
- b) $\dfrac{45}{105}$
- c) $\dfrac{256}{512}$
From the following list of fractions, group those that are equivalent to each other:…
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.
Finding a Missing Number
Sometimes an equivalent fraction has a blank, like $\frac{3}{5} = \frac{\Box}{20}$ or $\frac{3}{7} = \frac{21}{x}$. You have two tools.
Tool 1: look for the multiplier. Compare the two denominators: $5 \cdot 4 = 20$, so the numerator must also be multiplied by $4$. Then $\Box = 3 \cdot 4 = 12$.
Tool 2: cross products. When the multiplier is not obvious, use $a \cdot d = b \cdot c$ and solve.
The unknown can be inside an expression. Simplify the fraction you know first; it makes the numbers smaller.
Find the value of the missing number so that the following pair of fractions are equivalent: $$\frac{3}{5} = \frac{\Box}{20}$$
Find the value of the missing number so that the following pair of fractions are equivalent:
$$\frac{3}{5} = \frac{\Box}{20}$$Find the value of the unknown $x$ so that the following pairs of fractions are equivalent: a) $\dfrac{x}{15} = \dfrac{4}{5}$ b) $\dfrac{3}{7} = \dfrac{21}{x}$ c)…
Find the value of the unknown $x$ so that the following pairs of fractions are equivalent:
- a) $\dfrac{x}{15} = \dfrac{4}{5}$
- b) $\dfrac{3}{7} = \dfrac{21}{x}$
- c) $\dfrac{18}{x} = \dfrac{6}{11}$
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: a)…
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:
- a) $\frac{n+2}{12} = \frac{15}{18}$
- b) $\frac{8}{n-3} = \frac{32}{40}$
- c) $\frac{3}{7} = \frac{n^2}{147}$
Common Denominators and Different Wholes
Two fractions are easiest to compare when they have the same denominator. The best choice is the least common multiple (LCM) of the denominators, because it keeps the numbers small.
Equivalent fractions can also be used to check a claim. Irene's robot has finished $\frac{18}{24} = \frac{3}{4}$ of a track and Alexander's has finished $\frac{25}{30} = \frac{5}{6}$. With denominator $12$, those are $\frac{9}{12}$ and $\frac{10}{12}$. Alexander is ahead by $\frac{1}{12}$ of the track. (Notice that if Irene gains exactly that much, the two fractions become equal.)
A fraction of a quantity
To find a fraction of an amount, divide by the denominator and multiply by the numerator. If a $300$ ft² garden is split so Marcus gets $\frac{4}{15}$, then $300 \div 15 = 20$ ft² per fifteenth, and $20 \cdot 4 = 80$ ft².
Same fraction, different wholes
Back to the start: Marcus ($\frac{3}{4}$) and Lucy ($\frac{9}{12}$) have read the same fraction of equal-size books, because $3 \cdot 12 = 4 \cdot 9$. They are exactly equally far along.
In a robotics competition, Irene's robot has completed $\frac{18}{24}$ of the circuit, while Alexander's has completed $\frac{25}{30}$. a) Simplify both fractions to…
In a robotics competition, Irene's robot has completed $\frac{18}{24}$ of the circuit, while Alexander's has completed $\frac{25}{30}$.
- a) Simplify both fractions to their irreducible fraction to find out which robot is in the lead.
- 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.
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. a) Prove…
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.
- a) Prove graphically and using cross products whether the filling fractions are equivalent.
- 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".
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…
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.
- a) Find the simplified fraction that corresponds to Sophia.
- b) Find a fraction equivalent to Sophia's part whose numerator is double the numerator of Marcus's part.
- 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.