Rates of change and gradients of curves (Higher): แบบฝึกหัด

A freelancer's pay is increased by 20%. After six months, her new pay is decreased by 20%.
What is the overall percentage change in her pay from the original amount?

Two mathematically similar cylinders, \(A\) and \(B\), have heights in the ratio 2 : 5.
The volume of cylinder \(A\) is 160 cm\(^3\).
Calculate the volume of cylinder \(B\).

A metal bar has a mass of 780 g and a volume of 120 cm\(^3\).
Calculate the density of the metal bar.

\(P\) is inversely proportional to the square root of \(Q\).
When \(Q = 36\), \(P = 2\).
Find the value of \(P\) when \(Q = 16\).

A map has a scale of 1 : 25,000.
The distance between two villages on the map is 6.4 cm.
Calculate the actual distance between the two villages in kilometres.

A car's value, \(V\), depreciates such that its value after \(t\) years is given by \(V = A \times k^t\). The car was worth \(\text{\pounds}20,000\) when new and \(\text{\pounds}12,800\) after 2 years. Find the value of the car after 5 years.

In a sale, the price of a laptop is reduced by 15%. A week later, the sale price is reduced by a further 10%. The final price is \(\text{\pounds}459\). Calculate the original price of the laptop.

The variable \(z\) is directly proportional to the square of \(x\) and inversely proportional to \(y\). When \(x = 4\) and \(y = 8\), \(z = 10\). Find the value of \(z\) when \(x = 6\) and \(y = 3\).

A school holds a charity event. The ratio of the number of adults to the number of children at the event is 3 : 5.

(a) What fraction of the people at the event are children?

There are 120 people at the event in total.
(b) Work out the number of adults at the event.

An investment of \( \pounds 5000 \) is made in a savings account that pays \( 3.5\% \) compound interest per year.

(a) Calculate the value of the investment after 4 years. Give your answer to 2 decimal places.
(b) Find the number of years it takes for the investment to first exceed \( \pounds 7000 \).