In a game park \(\frac{1}{5}\) of the animals are rhinos and   \(\frac{3}{4}\) of them are zebras. Two-thirds of the remaining animals are lions and the rest are warthogs. Find the fraction of warthogs in the game park.

\(Solution\)

Let \(x\) represent the number of animals in the game park

 Rhinos 

\( = \frac{1}{5}x\)

 Zebras 

\(  =  \frac{3}{4}x\) 

 Remaining 

\( = x-(\frac{1}{5}x+ \frac{3}{4}x )\) 

\(=\frac{1}{20}x\)

 Lions 

\(  =  \frac{2}{3} of \frac{1}{20}x\) 

\(=\frac{1}{30}x \)

 Warthogs 

\(  =  \frac{1}{20}x -  \frac{1}{30}x  \)

\(=\frac{1}{60}x \) 

2. The average mark scored by the first 27 students in a mathematics test is 52. The average mark scored by the remaining 37 is 58. Calculate the mean mark for the whole class

average marks = \(\frac{27 \times 52+37\times 58}{27+37}\)

 \(=55.46875\) 

3. Use square roots, reciprocal and square tables to evaluate to 4 significant figures the expression

\(0.06458^{\frac{1}{2}} + (\frac{2}{0.4327})^{2}\)

It is advisable that we solve the first part of this expression. 

\((6.458 \times 10^{-2})^{\frac{1}{2}}\)

Applying the law of indices,  \((a^{m})^{n} = a^{m \times n}\), we multiply the powers and obtain:

\(6.458^{\frac{1}{2}} \times 10^{-1}\)

Remember, a number raised to the power of a half means the square root of that number

Now we get the square root of 6.458 from square roots table, therefore we have:

\(2.541 \times 10^{-1}\)

= 0.2541

For the second part, we need to rearrange the fraction within the bracket to be to apply table of reciprocals

Here is the rearrangement:

\(2 \times \frac{1}{0.4327}\)

Now we just use reciprocals table to solve  \(\frac{1}{0.4327} \). We need to rewrite the denominator in standard form

Therefore, we have:

\(2 \times  \frac{1}{4.327 \times 10^{-1}}\) 

This becomes   \(2 \times 0.2311 \times 10\)  which gives \(4.622\)

Remember we had the fraction inside the bracket raised to the power of 2 so we have \((4.622)^{2}\)   \(=21.36\) 

Therefore, recombining the two parts,

\(=0.2541 + 21.36\)

\(=21.6141\)