To solve this problem step-by-step, we need to follow the proportions and distributions mentioned:
1. Determine the shares of Alvin and Simon:
- The total amount of money is £540.
- The ratio in which Alvin and Simon share the money is 4:5.
- First, calculate the sum of the ratio parts: [tex]\( 4 + 5 = 9 \)[/tex] parts.
2. Calculate Alvin's share:
- Alvin receives [tex]\(\frac{4}{9}\)[/tex] of the total amount.
- Alvin's share = [tex]\( \frac{4}{9} \times 540 = 240 \)[/tex]
3. Calculate Simon's share:
- Simon receives [tex]\(\frac{5}{9}\)[/tex] of the total amount.
- Simon's share = [tex]\( \frac{5}{9} \times 540 = 300 \)[/tex]
4. Determine the amounts given to Theo:
- Alvin gave half of his share to Theo.
- Amount given by Alvin to Theo = [tex]\( \frac{1}{2} \times 240 = 120 \)[/tex]
- Simon gave a tenth of his share to Theo.
- Amount given by Simon to Theo = [tex]\( \frac{1}{10} \times 300 = 30 \)[/tex]
5. Calculate the total amount received by Theo:
- Total amount received by Theo = Amount from Alvin + Amount from Simon
- Total amount received by Theo = [tex]\( 120 + 30 = 150 \)[/tex]
6. Determine the fraction of the total money that Theo received:
- Fraction = [tex]\(\frac{\text{Total amount received by Theo}}{\text{Total amount of money}}
- Fraction = \(\frac{150}{540} = 0.2777\)[/tex]
This fraction can also be expressed as approximately 27.78%.
Thus, Theo received approximately [tex]\( \frac{5}{18} \)[/tex] or about 27.78% of the total £540.
