Operations with Rational Numbers
Regina records a video for her channel. While editing, she cuts $\frac{1}{6}$ of the footage because of sound errors, and then removes $\frac{2}{5}$ of what was left because the pacing was slow. What fraction of the original video survives? To answer, you need to add, subtract, multiply, and divide fractions with ease, even when negatives, powers, or decimals show up.
Simplest form of a fraction
A fraction is in simplest form (irreducible) when the numerator and the denominator no longer share any divisor other than $1$. To get there, you divide the numerator and denominator by their GCF, or by successive common divisors.
It's a good idea to simplify at the end of each operation, and sometimes beforehand, to keep the numbers small.
In a survey conducted in your class about their favorite social network, $\frac{12}{30}$ of the students chose a short-video platform. Find the irreducible fraction to…
In a survey conducted in your class about their favorite social network, $\frac{12}{30}$ of the students chose a short-video platform. Find the irreducible fraction to work out what fraction of the total this group represents.
Adding and subtracting fractions
To make a healthy smoothie, you have mixed $\frac{3}{4}$ of a liter of oat milk with $\frac{1}{2}$ liters of orange juice. When serving it, $\frac{1}{8}$ of a liter is…
To make a healthy smoothie, you have mixed $\frac{3}{4}$ of a liter of oat milk with $\frac{1}{2}$ liters of orange juice. When serving it, $\frac{1}{8}$ of a liter is spilled. Work through the operation step by step to find out how much of the mixture remains in the pitcher.
A teenager spends $\frac{1}{3}$ of their afternoon free time playing sports and $\frac{2}{5}$ studying math. a) Calculate what fraction of their total free time these…
A teenager spends $\frac{1}{3}$ of their afternoon free time playing sports and $\frac{2}{5}$ studying math.
- a) Calculate what fraction of their total free time these two activities take up.
- b) What fraction of time do they have left to play on the game console?
Multiplying and dividing fractions
To find a fraction of a quantity, multiply. For example, $\frac{4}{9}$ of $4{,}500$ mAh is $4{,}500\times\frac{4}{9}=\frac{18{,}000}{9}=2{,}000$ mAh. If you're looking for the whole when you know the part, you divide by the fraction.
On a shelf in a comic shop, $\frac{2}{3}$ of the titles are about superheroes. Of those superhero comics, half ($\frac{1}{2}$) belong to a famous American publisher.…
On a shelf in a comic shop, $\frac{2}{3}$ of the titles are about superheroes. Of those superhero comics, half ($\frac{1}{2}$) belong to a famous American publisher. Calculate the fraction of the whole shelf represented by the comics of that specific publisher by carrying out a fraction multiplication.
Your cell phone battery has a total capacity of $4500$ mAh. If the screen currently shows that exactly $\frac{4}{9}$ of its capacity remains: a) How many mAh are…
Your cell phone battery has a total capacity of $4500$ mAh. If the screen currently shows that exactly $\frac{4}{9}$ of its capacity remains:
- a) How many mAh are available?
- b) If battery saver mode is activated when $\frac{1}{5}$ of the total remains, how many mAh does that limit represent?
We want to share $\frac{3}{2}$ of a kilogram of gummy sweets into small bags. If we put $\frac{1}{10}$ of a kilogram into each bag: a) How many bags can we fill…
We want to share $\frac{3}{2}$ of a kilogram of gummy sweets into small bags. If we put $\frac{1}{10}$ of a kilogram into each bag:
- a) How many bags can we fill completely?
- b) Carry out the operation using division of fractions: $\frac{3}{2} : \frac{1}{10}$
Opposite and reciprocal of a fraction
Look at the following figure, which represents a chocolate cake divided into equal slices. Write down the fraction representing the missing part, and calculate the…
Look at the following figure, which represents a chocolate cake divided into equal slices. Write down the fraction representing the missing part, and calculate the opposite fraction and the reciprocal fraction of that amount.
Order of operations with fractions
The hierarchy is the same as with integers: first grouping symbols (parentheses, brackets), then powers and roots, then multiplications and divisions from left to right, and finally additions and subtractions.
Solve the following mixed operation respecting the order of operations: $$\frac{2}{5} \times \left( \frac{1}{2} + \frac{1}{3} \right) - \frac{1}{10}$$
Solve the following mixed operation respecting the order of operations:
$$\frac{2}{5} \times \left( \frac{1}{2} + \frac{1}{3} \right) - \frac{1}{10}$$Solve the following combined expression, simplifying the result to its simplest form: $$\frac{3}{5} - \left[ \frac{1}{2} : \left( \frac{2}{3} - \frac{1}{6} \right)…
Solve the following combined expression, simplifying the result to its simplest form:
$$\frac{3}{5} - \left[ \frac{1}{2} : \left( \frac{2}{3} - \frac{1}{6} \right) \right] \times \frac{5}{4}$$Calculate the result of the following combined operation, simplifying the result to its simplest form: $$ \frac{3}{5} - \left[ \frac{1}{2} + \left( \frac{2}{3} \right)^2…
Calculate the result of the following combined operation, simplifying the result to its simplest form:
$$ \frac{3}{5} - \left[ \frac{1}{2} + \left( \frac{2}{3} \right)^2 \times \left( \frac{1}{4} - 1 \right) \right] $$Powers and roots with fractions
Powers of a fraction are calculated by raising the numerator and denominator: $\left(\frac{a}{b}\right)^n=\frac{a^n}{b^n}$. With a negative exponent, the fraction is flipped.
Calculate the value of the following mixed operation with negative powers and simplify the result: $$ \left( \frac{3}{2} \right)^{-2} : \left( \frac{1}{2} - \frac{1}{3}…
Calculate the value of the following mixed operation with negative powers and simplify the result:
$$ \left( \frac{3}{2} \right)^{-2} : \left( \frac{1}{2} - \frac{1}{3} \right) + \left( -1 \right)^5 $$Find the value of the following expression taking into account the order of operations and the sign rules: $$ \frac{\frac{3}{2} - \frac{5}{4}}{\frac{1}{2} \times \left(…
Find the value of the following expression taking into account the order of operations and the sign rules:
$$ \frac{\frac{3}{2} - \frac{5}{4}}{\frac{1}{2} \times \left( -3 \right)} + \sqrt{\frac{16}{81}} $$Find the value of $x$ in the following equality of powers with rational numbers, applying the necessary properties to simplify the expression: $$\frac{\left( \frac{2}{3}…
Find the value of $x$ in the following equality of powers with rational numbers, applying the necessary properties to simplify the expression:
$$\frac{\left( \frac{2}{3} \right)^5 \times \left( \frac{9}{4} \right)^{-2}}{\left( \frac{4}{9} \right)^{3} : \left( \frac{3}{2} \right)^{-4}} = \left( \frac{2}{3} \right)^x$$Decimals within operations
When an expression mixes fractions and decimals, convert the decimals to fractions and work only with fractions. That way the result is exact.
In a chemistry laboratory, a container holds $2.4$ liters of an acid solution. For an experiment, $\frac{2}{3}$ of the contents is first extracted. Then, $0.5$ liters of…
In a chemistry laboratory, a container holds $2.4$ liters of an acid solution. For an experiment, $\frac{2}{3}$ of the contents is first extracted. Then, $0.5$ liters of distilled water are added to dilute the mixture. Finally, $\frac{3}{5}$ of the resulting mixture is used.
- a) Carry out all operations working exclusively with fractions.
- b) How many liters of mixture remain in the container at the end? Express the result both as a fraction and as a decimal number.
Calculate the exact value of the following combined expression, first converting the decimals into their respective generating fractions and simplifying the intermediate…
Calculate the exact value of the following combined expression, first converting the decimals into their respective generating fractions and simplifying the intermediate steps:
$$\left( \frac{3}{5} - 0,\widehat{6} \right)^2 \times \left( \frac{1}{3} \right)^{-1} + \sqrt{\frac{25}{9}} : 1.2\widehat{5}$$"Of the remainder" problems with fractions
When a problem says a fraction of what's left, that fraction applies to the remainder, not to the total. Calculate stage by stage.
If you're given what's left and asked for the total, you divide that amount by the fraction it represents.
Back to the beginning: of Regina's video, $\frac{5}{6}-\frac{1}{3}=\frac{1}{2}$ survives. If the final video lasts 18 minutes, the original recording lasted $18\div\frac{1}{2}=36$ minutes.
Three friends decide to buy a gift for a classmate who is changing schools. The first contributes $\frac{2}{5}$ of the total price, the second contributes $\frac{1}{3}$…
Three friends decide to buy a gift for a classmate who is changing schools. The first contributes $\frac{2}{5}$ of the total price, the second contributes $\frac{1}{3}$ of what is left, and the third completes the payment with $320$ pesos.
- a) What fraction of the total does the amount contributed by the second friend represent?
- b) What is the total price of the gift?
- c) How much money did each person put in?
A content creator recorded a video for their channel. During editing, they cut $\frac{1}{6}$ of the total footage due to sound errors. Then, they removed $\frac{2}{5}$…
A content creator recorded a video for their channel. During editing, they cut $\frac{1}{6}$ of the total footage due to sound errors. Then, they removed $\frac{2}{5}$ of what was left because the pacing was too slow. In the end, the edited video is $18$ minutes long.
- a) What fraction of the original footage does the final $18$-minute video represent?
- b) How long was the original recording before editing began?
A young influencer is organising her monthly budget from sponsorships. She allocates $\frac{2}{5}$ of her income to renting her studio and $\frac{1}{4}$ of the remainder…
A young influencer is organising her monthly budget from sponsorships. She allocates $\frac{2}{5}$ of her income to renting her studio and $\frac{1}{4}$ of the remainder to video editing. If after these expenses she still sets aside $\frac{5}{9}$ of what was left for savings and still has $9\,600$ pesos left for personal expenses, what is her total monthly income?
Fractions of a whole and of a figure
The parts of a whole add up to $1$ (the whole). If you know some of the parts, the rest is found by subtracting: $1-\ldots$
If the figure is divided into equal squares or triangles, count how many equal parts fit in the whole and how many are shaded. If there are nested regions, calculate each region as a fraction of the previous one and multiply.
An architect is designing a play area in a rectangular park. They have decided to divide the ground as shown in the following diagram: If the "Grass" area occupies…
An architect is designing a play area in a rectangular park. They have decided to divide the ground as shown in the following diagram:
If the "Grass" area occupies exactly half of the ground and the "Swings" area is $\frac{3}{10}$ of the total park:
- a) What fraction of the total does the "Sand" area represent?
- b) If the park has a total area of $1200$ $m^2$, how many square meters is the "Sand" area?
Look at the design of this modern tile for a cultural center. The outer square has an area of $1$ m$^2$. Different regions have been shaded following a geometric…
Look at the design of this modern tile for a cultural center. The outer square has an area of $1$ m$^2$. Different regions have been shaded following a geometric pattern. Determine what fraction of the total the orange shaded region represents and calculate how many cm$^2$ correspond to it.
Multi-step problems and equations
In longer problems it helps to call the total $T$ and express each part as a fraction of $T$. Then you set up an equation and solve it by multiplying by the LCM of the denominators.
When comparing speeds and times over the same route, remember that they are reciprocals: if the speed drops to $\frac{2}{3}$, the time is multiplied by $\frac{3}{2}$.
In an eSports competition, the total prize money is distributed as follows: the winning team gets $\frac{4}{7}$ of the total, the runner-up $\frac{2}{3}$ of what is…
In an eSports competition, the total prize money is distributed as follows: the winning team gets $\frac{4}{7}$ of the total, the runner-up $\frac{2}{3}$ of what is left, and the third-placed team the remainder. If the difference between the prize for second and third place is $70\,000$ pesos, calculate the total value of the prize money and how much money each of the three teams receives.
A surveillance drone makes a round trip between two bases. On the outward journey, it flies with a tailwind at a constant speed. On the return journey, into a headwind,…
A surveillance drone makes a round trip between two bases. On the outward journey, it flies with a tailwind at a constant speed. On the return journey, into a headwind, its speed is reduced to $\frac{3}{5}$ of the outward speed. If the total flight time was $1$ hour and $20$ minutes, what fraction of the total time did it take for the return journey? Express the result in minutes and seconds as well.
Three partners invest in a technology business. The first contributes $240\,000$ pesos, the second contributes $\frac{2}{5}$ of the total, and the third contributes the…
Three partners invest in a technology business. The first contributes $240\,000$ pesos, the second contributes $\frac{2}{5}$ of the total, and the third contributes the same amount as the other two combined minus $360\,000$ pesos.
- a) What is the total investment in the business?
- b) What fraction of the total does each partner contribute? (Express the fractions in their simplest form).
In a chemistry laboratory, a tank contains a mixture of alcohol and water. Initially, $\frac{2}{7}$ of the contents is alcohol. $12$ liters of the mixture are removed…
In a chemistry laboratory, a tank contains a mixture of alcohol and water. Initially, $\frac{2}{7}$ of the contents is alcohol. $12$ liters of the mixture are removed and replaced with $12$ liters of pure alcohol. If the mixture now contains $\frac{2}{5}$ alcohol, how many liters of liquid are in total in the tank?