1. **Problem Statement:** We have income and expenditure data for companies ABC and DEF over years 2001-2006, with some ratios and values given. We need to find various ratios, incomes, expenditures, total profits, and profit percentages. 2. **Given Formula:** Profit percent = $$\frac{\text{Income} - \text{Expenditure}}{\text{Expenditure}} \times 100$$ 3. **Step-by-step Solutions:** **(a) Ratio of expenditures of ABC and DEF in 2005 given income ratio 3:4** - Let income of ABC in 2005 = $3x$, income of DEF in 2005 = $4x$ - From the graph, profit percentages for 2005 are approximately: - ABC: 60% - DEF: 50% - Using profit percent formula: $$60 = \frac{3x - E_{ABC}}{E_{ABC}} \times 100 \Rightarrow 0.6 = \frac{3x - E_{ABC}}{E_{ABC}}$$ $$0.6 E_{ABC} = 3x - E_{ABC} \Rightarrow 1.6 E_{ABC} = 3x \Rightarrow E_{ABC} = \frac{3x}{1.6}$$ $$50 = \frac{4x - E_{DEF}}{E_{DEF}} \times 100 \Rightarrow 0.5 = \frac{4x - E_{DEF}}{E_{DEF}}$$ $$0.5 E_{DEF} = 4x - E_{DEF} \Rightarrow 1.5 E_{DEF} = 4x \Rightarrow E_{DEF} = \frac{4x}{1.5}$$ - Ratio of expenditures: $$\frac{E_{ABC}}{E_{DEF}} = \frac{\frac{3x}{1.6}}{\frac{4x}{1.5}} = \frac{3}{1.6} \times \frac{1.5}{4} = \frac{3 \times 1.5}{1.6 \times 4} = \frac{4.5}{6.4} = \frac{45}{64}$$ **Answer (a):** Ratio of expenditures ABC : DEF = 45 : 64 **(b) Income of DEF in 2002 given expenditure = 190 cr.** - From graph, profit % of DEF in 2002 is about 40% - Using formula: $$40 = \frac{I - 190}{190} \times 100 \Rightarrow 0.4 = \frac{I - 190}{190}$$ $$0.4 \times 190 = I - 190 \Rightarrow 76 = I - 190 \Rightarrow I = 266$$ **Answer (b):** Income of DEF in 2002 = 266 cr. **(c) Total profit of ABC and DEF in 2001 given expenditures equal and total income 825 cr.** - Let expenditure of both companies in 2001 = $E$ - From graph, profit % in 2001: - ABC: 50% - DEF: 40% - Profit % formula: $$50 = \frac{I_{ABC} - E}{E} \times 100 \Rightarrow I_{ABC} = 1.5 E$$ $$40 = \frac{I_{DEF} - E}{E} \times 100 \Rightarrow I_{DEF} = 1.4 E$$ - Total income: $$I_{ABC} + I_{DEF} = 1.5 E + 1.4 E = 2.9 E = 825 \Rightarrow E = \frac{825}{2.9} = 284.48$$ - Total profit: $$(I_{ABC} - E) + (I_{DEF} - E) = (1.5E - E) + (1.4E - E) = 0.5E + 0.4E = 0.9E = 0.9 \times 284.48 = 256.03$$ **Answer (c):** Total profit in 2001 = 256.03 cr. **(d) Expenditure of ABC in 2004 given income = 750 cr.** - Profit % of ABC in 2004 from graph is about 55% - Using formula: $$55 = \frac{750 - E}{E} \times 100 \Rightarrow 0.55 = \frac{750 - E}{E}$$ $$0.55 E = 750 - E \Rightarrow 1.55 E = 750 \Rightarrow E = \frac{750}{1.55} = 483.87$$ **Answer (d):** Expenditure of ABC in 2004 = 483.87 cr. **(e) Ratio of expenditures of ABC and DEF in 2003 given incomes equal** - Let income of both companies in 2003 = $I$ - Profit % in 2003 from graph: - ABC: 45% - DEF: 65% - Using formula: $$45 = \frac{I - E_{ABC}}{E_{ABC}} \times 100 \Rightarrow 0.45 = \frac{I - E_{ABC}}{E_{ABC}} \Rightarrow 1.45 E_{ABC} = I$$ $$65 = \frac{I - E_{DEF}}{E_{DEF}} \times 100 \Rightarrow 0.65 = \frac{I - E_{DEF}}{E_{DEF}} \Rightarrow 1.65 E_{DEF} = I$$ - Since incomes equal: $$1.45 E_{ABC} = 1.65 E_{DEF} \Rightarrow \frac{E_{ABC}}{E_{DEF}} = \frac{1.65}{1.45} = \frac{165}{145} = \frac{33}{29}$$ **Answer (e):** Ratio of expenditures ABC : DEF = 33 : 29 **(f) Profit percent of ABC and DEF during 2001-2006** - From graph approximate profit % values: - ABC: 2001=50%, 2002=55%, 2003=45%, 2004=55%, 2005=60%, 2006=50% - DEF: 2001=40%, 2002=40%, 2003=65%, 2004=55%, 2005=50%, 2006=45% **Answer (f):** - ABC profit %: 50, 55, 45, 55, 60, 50 - DEF profit %: 40, 40, 65, 55, 50, 45 **Summary:** - (a) Expenditure ratio 2005: 45:64 - (b) DEF income 2002: 266 cr. - (c) Total profit 2001: 256.03 cr. - (d) ABC expenditure 2004: 483.87 cr. - (e) Expenditure ratio 2003: 33:29 - (f) Profit % 2001-2006 as above