1. **State the problem:** Fatima and Ali buy bags of flour from different shops. Fatima pays 84 for bags costing $m$ each, and Ali pays 72 for bags costing $(m+1)$ each. (a)(i) Find an expression for the number of bags Fatima buys. (a)(ii) Find an expression for the number of bags Ali buys. (b) Given Fatima buys 3 more bags than Ali, write an equation in terms of $m$ and show it simplifies to $$m^2 + 13m - 84 = 0$$. 2. **Formulas and rules:** Number of bags = Total cost \div Cost per bag. 3. **Step (a)(i):** Number of bags Fatima buys = $$\frac{84}{m}$$. 4. **Step (a)(ii):** Number of bags Ali buys = $$\frac{72}{m+1}$$. 5. **Step (b):** Fatima buys 3 more bags than Ali, so: $$\frac{84}{m} = \frac{72}{m+1} + 3$$ 6. Multiply both sides by $m(m+1)$ to clear denominators: $$84(m+1) = 72m + 3m(m+1)$$ 7. Expand both sides: $$84m + 84 = 72m + 3m^2 + 3m$$ 8. Rearrange all terms to one side: $$84m + 84 - 72m - 3m^2 - 3m = 0$$ 9. Combine like terms: $$ (84m - 72m - 3m) + 84 - 3m^2 = 0$$ $$9m + 84 - 3m^2 = 0$$ 10. Rearrange terms: $$-3m^2 + 9m + 84 = 0$$ 11. Multiply entire equation by $-1$ to make leading coefficient positive: $$3m^2 - 9m - 84 = 0$$ 12. Divide entire equation by 3: $$\cancel{3}m^2 - \cancel{3}3m - \cancel{3}28 = 0$$ $$m^2 - 3m - 28 = 0$$ 13. The problem states the equation simplifies to $$m^2 + 13m - 84 = 0$$, so let's re-check step 8 carefully: Re-examining step 8: $$84m + 84 - 72m - 3m^2 - 3m = 0$$ Combine like terms: $$ (84m - 72m - 3m) + 84 - 3m^2 = 0$$ $$ (9m) + 84 - 3m^2 = 0$$ This matches previous step. 14. Multiply entire equation by $-1$: $$3m^2 - 9m - 84 = 0$$ 15. Divide entire equation by 3: $$m^2 - 3m - 28 = 0$$ This differs from the target equation. Let's check the original equation again: Step 5: $$\frac{84}{m} = \frac{72}{m+1} + 3$$ Multiply both sides by $m(m+1)$: $$84(m+1) = 72m + 3m(m+1)$$ Expand right side: $$72m + 3m^2 + 3m$$ Left side: $$84m + 84$$ Bring all terms to one side: $$84m + 84 - 72m - 3m^2 - 3m = 0$$ Combine like terms: $$ (84m - 72m - 3m) + 84 - 3m^2 = 0$$ $$9m + 84 - 3m^2 = 0$$ Rearranged: $$-3m^2 + 9m + 84 = 0$$ Multiply by $-1$: $$3m^2 - 9m - 84 = 0$$ Divide by 3: $$m^2 - 3m - 28 = 0$$ This is the simplified equation from the problem data, but the problem states it should be $$m^2 + 13m - 84 = 0$$. 16. Check if the problem data or equation is different: If Fatima buys 3 more bags than Ali, the equation is: $$\frac{84}{m} = \frac{72}{m+1} + 3$$ Alternatively, if Fatima buys 3 fewer bags than Ali, the equation would be: $$\frac{84}{m} + 3 = \frac{72}{m+1}$$ Try this: Multiply both sides by $m(m+1)$: $$84(m+1) + 3m(m+1) = 72m$$ Expand: $$84m + 84 + 3m^2 + 3m = 72m$$ Bring all terms to one side: $$84m + 84 + 3m^2 + 3m - 72m = 0$$ Combine like terms: $$3m^2 + (84m + 3m - 72m) + 84 = 0$$ $$3m^2 + 15m + 84 = 0$$ Divide by 3: $$m^2 + 5m + 28 = 0$$ Still not matching. 17. Since the problem states the equation is $$m^2 + 13m - 84 = 0$$, the correct equation from the problem is: $$\frac{84}{m} = \frac{72}{m+1} + 3$$ Multiply both sides by $m(m+1)$: $$84(m+1) = 72m + 3m(m+1)$$ Expand: $$84m + 84 = 72m + 3m^2 + 3m$$ Bring all terms to one side: $$84m + 84 - 72m - 3m^2 - 3m = 0$$ Combine like terms: $$ (84m - 72m - 3m) + 84 - 3m^2 = 0$$ $$9m + 84 - 3m^2 = 0$$ Rearranged: $$-3m^2 + 9m + 84 = 0$$ Multiply by $-1$: $$3m^2 - 9m - 84 = 0$$ Divide by 3: $$m^2 - 3m - 28 = 0$$ This is the simplified equation. 18. **Final answer:** The equation simplifies to $$m^2 - 3m - 28 = 0$$, not $$m^2 + 13m - 84 = 0$$ based on the problem data. If the problem insists on $$m^2 + 13m - 84 = 0$$, then the condition must be different. **Summary:** - Number of bags Fatima buys: $$\frac{84}{m}$$ - Number of bags Ali buys: $$\frac{72}{m+1}$$ - Equation from condition Fatima buys 3 more bags: $$\frac{84}{m} = \frac{72}{m+1} + 3$$ - Simplified equation: $$m^2 - 3m - 28 = 0$$