Operations with rational numbers
You can already add, subtract, multiply and divide fractions separately. Here you will use them inside real multi-step problems: a half-sized recipe, a cyclist riding in stages, an unknown hidden in an equality with fractions.
A quick review of combined operations
Work out the following combined operation with fractions, respecting the order of operations, and express the result as an irreducible fraction:…
Work out the following combined operation with fractions, respecting the order of operations, and express the result as an irreducible fraction:
$ \frac{3}{4} - \left( \frac{1}{2} + \frac{2}{5} \right) \cdot \frac{5}{3} $
A fraction of a fraction
"Half of three quarters" does not mean adding, it means multiplying: whenever you read "of" between two fractions or quantities, it is a multiplication.
A baking recipe requires $\frac{3}{4}$ of a kilogram of flour to make a whole cake. If we only have enough ingredients to make half the recipe, how much flour will we…
A baking recipe requires $\frac{3}{4}$ of a kilogram of flour to make a whole cake. If we only have enough ingredients to make half the recipe, how much flour will we need? Express the result as an irreducible fraction.
Compound fractions: the fraction tower
A fraction can have another fraction in its numerator or denominator. It is solved from the inside out, like any combined operation: first the top, then the bottom, and finally you divide.
Solve the following complex fraction, simplifying the numerator and the denominator before performing the main division: $ \frac{1 + \frac{1}{2}}{1 - \frac{1}{3}} $
Solve the following complex fraction, simplifying the numerator and the denominator before performing the main division:
$ \frac{1 + \frac{1}{2}}{1 - \frac{1}{3}} $
The unknown inside a fraction
Find the value of the unknown $x$ that makes the following equation true: $ \frac{2}{3} \cdot x - \frac{1}{4} = \frac{5}{12} $
Find the value of the unknown $x$ that makes the following equation true:
$ \frac{2}{3} \cdot x - \frac{1}{4} = \frac{5}{12} $
"What is left" problems
The challenge is not the calculation, but reading carefully which quantity each fraction applies to: sometimes it is the total, and sometimes it is "what is left" after another share.
A cyclist completes $\frac{1}{3}$ of a stage in the first hour. In the second hour, he covers $\frac{2}{5}$ of the remaining distance. If after these two hours he still…
A cyclist completes $\frac{1}{3}$ of a stage in the first hour. In the second hour, he covers $\frac{2}{5}$ of the remaining distance. If after these two hours he still has 30 km left to reach the finish line, what is the total length of the stage?
A technology company allocates $\frac{3}{8}$ of its annual budget to salaries, $\frac{1}{5}$ to marketing campaigns and $\frac{1}{4}$ of the remaining budget to research…
A technology company allocates $\frac{3}{8}$ of its annual budget to salaries, $\frac{1}{5}$ to marketing campaigns and $\frac{1}{4}$ of the remaining budget to research and development (R&D). What fraction of the total budget remains available for other operating expenses?
A metal alloy used in aerospace engineering consists of $\frac{3}{7}$ copper, $\frac{2}{5}$ zinc, and the rest is tin. If the total mass of a part made from this alloy…
A metal alloy used in aerospace engineering consists of $\frac{3}{7}$ copper, $\frac{2}{5}$ zinc, and the rest is tin. If the total mass of a part made from this alloy is 140 kg, how many exact kilograms of tin does it contain?
One tap can fill a swimming pool in 4 hours, while a second tap can fill it in 6 hours. If both taps are opened at the same time, what fraction of the pool will be…
One tap can fill a swimming pool in 4 hours, while a second tap can fill it in 6 hours. If both taps are opened at the same time, what fraction of the pool will be filled after 1 hour? How much total time (in hours and minutes) will they take to fill the pool completely?
Work out the exact sum of the following series of fractions. (Hint: Observe the pattern and split each fraction into the difference of two simpler fractions, for example…
Work out the exact sum of the following series of fractions. (Hint: Observe the pattern and split each fraction into the difference of two simpler fractions, for example $\frac{1}{n \cdot (n+1)} = \frac{1}{n} - \frac{1}{n+1}$):
$ \frac{1}{1 \cdot 2} + \frac{1}{2 \cdot 3} + \frac{1}{3 \cdot 4} + \dots + \frac{1}{9 \cdot 10} $