Exercise solution 8, page 185 - GCSE Mathematics for OCR Higher Student Book

⭐ What you need to know:

A) A mixed number is a whole number together with a fraction. For example, \(5\frac{3}{8}\) means \(5+\frac{3}{8}\). To work more easily, it is often useful to change a mixed number into an improper fraction.

\[ \begin{aligned} 5\frac{3}{8} &= \frac{5\times 8}{8}+\frac{3}{8} \\ &= \frac{40}{8}+\frac{3}{8} \\ &= \frac{43}{8} \end{aligned} \]

B) A fraction of a quantity means “that part of the whole”. To find a fraction of a length, multiply the total length by the fraction. For instance, if \(\frac{7}{20}\) of a pole is above the ground and the pole is \(9\) m long, then the part above the ground is:

\[ \begin{aligned} \frac{7}{20}\times 9 &= \frac{7}{20}\times \frac{9}{1} \\ &= \frac{63}{20} \\ &= 3.15 \end{aligned} \]

So the visible part is \(3.15\) m.

C) If one part of a length is known, the remaining part can be found by subtraction. In this type of question, the ladder is split into two parts: the part outside the manhole and the part inside the manhole. Since the whole ladder length is known, the depth of the manhole is found by:

depth = total length of ladder - length outside

For example, if a rod is \(8.4\) m long and \(2.1\) m remains above a hole, then the part inside the hole is:

\[ \begin{aligned} 8.4 - 2.1 &= 6.3 \end{aligned} \]

So the hole is \(6.3\) m deep.

D) When multiplying fractions, multiply the numerators together and the denominators together. If needed, simplify the result or convert it into a decimal or mixed number.

\[ \begin{aligned} \frac{4}{9}\times \frac{15}{2} &= \frac{4\times 15}{9\times 2} \\ &= \frac{60}{18} \\ &= \frac{10}{3} \\ &= 3\frac{1}{3} \end{aligned} \]

E) Always check that the answer is sensible. The depth of the manhole must be less than the full length of the ladder, because part of the ladder is still outside. So if the ladder is \(7\) m long, the depth must be < \(7\) m. A quick estimate can help: if about half the ladder is outside, then about half is inside as well.