Bat and Ball

Answer: £60

Using the cost of 3 balls and 3 bats 

If one ball and one bat cost £90, then three balls and three bats cost £90$\times$3 = £270.

Three balls and two bats cost £210, and £210 + £60 = £270.

So the extra bat must cost £60.

Using simultaneous equations

Let a bat cost £$a$ and a ball cost £$b$. Then $a + b = 90$, and $3b + 2a = 210$.

Solving by elimination

Looking for $a$ so eliminate $b$

Multiply both sides by $3$:

$a+b=90\Rightarrow 3a+3b=270$.

Subtracting $3b+2a=210$ from $3a+3b=270$ gives $3a+3b-(3b+2a)=270-210,$ so $3a+3b-3b-2a=60,$ so $a=60.$ So a bat costs £$60.$

Solving by substitution

Looking for $a$ so express $b$ in terms of $a$

$a+b=90\Rightarrow b=90-a.$ Substituting this into $3b + 2a = 210$ gives $3(90-a)+2a=210.$ Expanding, simplifying and rearranging, $$\begin{align}3(90-a)+2a=&210

\\\Rightarrow 270-3a+2a=&210\\

\Rightarrow270-a=&210\\

\Rightarrow 60=&a\end{align}$$ So a bat costs £$60$.