# Maths Week teaser

*Mary has twice as many brothers as sisters while her brother John has twice as many sisters as brothers. How many children are in their family?*

Answer: four children

Mary has 2 brothers and one sister

John has 2 sisters and 1 brother

This sort of puzzle can be solved using a technique called **simultaneous equations.**

This is a technique that is covered in Junior Certificate mathematics and is very powerful in finding unknown quantities in a system if one has enough information to piece together mathematically. It's not really necessary for the above question that may be solved by informed "trial and error" but we will take the opportunity of showing the technique here. (We are also running a Simultaneous Simultaneous Equations activity see here)

*We have two related pieces of information:*

*1. Mary has twice as many brothers and sisters*

*2. John has twice as many brothers as sisters*

*We can express each of these with symbols and numbers. We use symbols as a shorthand code.*

*Here we will use the letter B to represent the number of boys*

*and the letter G to represent the number of girls*

*1. " Mary has twice as many brothers and sisters" can be expressed symbolically. The number of sisters Mary has is G-1. This is because Mary is a girl and therefore her number of sisters is the number of girls in the family minus one (herself). The number of brothers is the number of boys (B) and this is twice the number of sisters (2x(G-1))*

*B =2(G-1) *

*which multiplies out as*

* **B = 2G - 2 (which we will call equation 1(eqn 1))*

*Likewise *

*2. "John has twice as many sisters as brothers"** can be expressed as*

*G = 2(B-1) *

*which multiplies out as*

* **G = 2B - 2 (which we will call equation 2(eqn 2))*

*Because both of these mathematical equations describe the same situation they can be combined. *

If we take (eqn 1) *B =2G-2** and **insert this expression for B into (eqn 2) G = 2B-2 we get:*

*G = 2(2G-2) - 2 *

*Whish multiplies out as *

*G = 4G - 4 -2*

*which is *

*G = 4G -6*

*if we subtract G from both sides*

*0 = 3G - 6*

*and if we add 6 to each side we get*

*3G = 6*

*And if 3G = 6 then G must equal 2. *

*Therefore the number of girls is 2. *

*Going back to eqn 1*

* B = 2G - 2 and putting in G=2*

*B = 2(2) - 2*

*B = 4 - 2*

*B = 2*

*So there are two boys and two girls in family. *