Question: Daniel shares £430 between charities A, B and C.
The ratio of the money that A receives to the money that B receives is 5 : 2
The ratio of the money that A receives to the money that C receives is 4 : 3
Work out how much money C receives.
1. **State the problem:** Daniel shares 430 between charities A, B, and C. The ratio of A to B is 5:2, and the ratio of A to C is 4:3. We need to find how much C receives.
2. **Express the ratios with variables:** Let the amounts received by A, B, and C be $A$, $B$, and $C$ respectively.
Given:
$$\frac{A}{B} = \frac{5}{2} \implies A = \frac{5}{2}B$$
$$\frac{A}{C} = \frac{4}{3} \implies A = \frac{4}{3}C$$
3. **Find a common expression for $A$:** From the two equations, express $B$ and $C$ in terms of $A$:
$$B = \frac{2}{5}A$$
$$C = \frac{3}{4}A$$
4. **Sum of all amounts:** The total amount shared is 430:
$$A + B + C = 430$$
Substitute $B$ and $C$:
$$A + \frac{2}{5}A + \frac{3}{4}A = 430$$
5. **Find common denominator and simplify:**
$$A + \frac{2}{5}A + \frac{3}{4}A = A\left(1 + \frac{2}{5} + \frac{3}{4}\right) = 430$$
Calculate inside the parentheses:
$$1 = \frac{20}{20}, \frac{2}{5} = \frac{8}{20}, \frac{3}{4} = \frac{15}{20}$$
So:
$$\frac{20}{20} + \frac{8}{20} + \frac{15}{20} = \frac{43}{20}$$
6. **Equation becomes:**
$$A \times \frac{43}{20} = 430$$
7. **Solve for $A$:**
$$A = 430 \times \frac{20}{43}$$
Simplify:
$$A = 10 \times 20 = 200$$
8. **Find $C$ using $A = \frac{4}{3}C$:**
$$C = \frac{3}{4}A = \frac{3}{4} \times 200 = 150$$
**Final answer:** C receives 150.