Rational numbers (fractions)

Simplify the following fraction to its simplest form: $\frac{1260}{1500}$.

Express the purely recurring decimal number $3,\overline{45}$ as an irreducible fraction.

Without performing the division, determine which of the following fractions will result in a terminating decimal and which in a recurring decimal, justifying your answer by factorising the denominators: $\frac{13}{40}$, $\frac{5}{21}$, $\frac{27}{250}$.

Look at the following number line. Identify which rational numbers (as irreducible fractions) correspond to the points marked with the letters A, B and C.

Work out the following combined calculation with fractions and simplify the result: $\frac{3}{4} - \frac{1}{2} \cdot \left( \frac{5}{6} + \frac{1}{3} \right)$.

An employee uses $\frac{1}{4}$ of his monthly salary to pay the rent for his home and $\frac{2}{5}$ of the remainder for food and utility expenses. If after making these payments he has $810$ euros left in his account, what is his total monthly salary?

A reservoir used for irrigation was filled to $\frac{7}{8}$ of its total capacity. After a week of severe drought, an amount of water equivalent to $\frac{1}{5}$ of the total capacity of the reservoir evaporates and is consumed. Later, heavy rainfall adds an amount of water equal to $\frac{1}{4}$ of the water remaining at that moment. If the total capacity of the reservoir is $1200$ cubic hectometres, what fraction of the total capacity of the reservoir is now full, and how many cubic hectometres does it contain?

A rubber ball special for physics experiments is dropped from an initial height of $10$ metres. After hitting the ground, on each bounce it reaches a height equal to $\frac{3}{4}$ of the height reached on the immediately preceding bounce. What height does the ball reach after the third bounce? Express the final result first as an irreducible fraction and then as a decimal number.

A clockwork mechanism consists of a system of three gears A, B and C connected in series (A drives B, and B drives C). Gear A has $12$ teeth, gear B has $18$ teeth and gear C has $30$ teeth. Knowing that the number of turns a gear makes is inversely proportional to its number of teeth, if gear A turns exactly $\frac{15}{4}$ of a turn, what fraction of a turn will gear C make?

In the architectural design of a mosaic, an initial square of area $1 \text{ m}^2$ is drawn. Inside it, a second square is inscribed by joining the midpoints of the sides of the first. This geometric process is repeated to draw a third square inscribed in the second, and a fourth square inscribed in the third. What fraction of the area of the first square represents the area contained exclusively between the third and fourth squares?

You can do it on paper and send a photo with your answer.