This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Here are the solutions for Question 13 and 14: 13. Calculate: i) 6(3)/(5) × 2(1)/(4) ÷ 3(1)/(2) + 5(1)/(2) Step 1: Convert all mixed numbers to improper fractions. 6(3)/(5) = (33)/(5) 2(1)/(4) = (9)/(4) 3(1)/(2) = (7)/(2) 5(1)/(2) = (11)/(2) The expression becomes: (33)/(5) × (9)/(4) ÷ (7)/(2) + (11)/(2) Step 2: Perform multiplication and division from left to right. First, multiply: (33)/(5) × (9)/(4) = (297)/(20) The expression is now: (297)/(20) ÷ (7)/(2) + (11)/(2) Next, divide (multiply by the reciprocal): (297)/(20) × (2)/(7) = (297 × 2)/(20 × 7) = (594)/(140) = (297)/(70) The expression is now: (297)/(70) + (11)/(2) Step 3: Perform addition. Find a common denominator, which is 70. (11)/(2) = (11 × 35)/(2 × 35) = (385)/(70) (297)/(70) + (385)/(70) = (297 + 385)/(70) = (682)/(70) Step 4: Simplify the fraction. (682)/(70) = (341)/(35) As a mixed number: 9(26)/(35) The calculated value is (341)/(35) or 9(26)/(35). ii) sqrt((5)/(8) × (3)/(4) ÷ (4)/((6)+2)) Step 1: Simplify the denominator of the last fraction. (6)+2 = 8 The last fraction is (4)/(8), which simplifies to (1)/(2). The expression becomes: sqrt((5)/(8) × (3)/(4) ÷ (1)/(2)) Step 2: Perform multiplication and division inside the square root from left to right. First, multiply: (5)/(8) × (3)/(4) = (15)/(32) The expression is now: sqrt((15)/(32) ÷ (1)/(2)) Next, divide (multiply by the reciprocal): (15)/(32) × (2)/(1) = (30)/(32) = (15)/(16) The expression is now: sqrt((15)/(16)) Step 3: Calculate the square root. sqrt((15)/(16)) = sqrt(15)sqrt(16) = sqrt(15)4 The calculated value is sqrt(15)4. 14. The sum of K1050 is to be shared among three children. The second born gets (1)/(3) of the share of the first born. The youngest gets (1)/(6) of the share of the first born. Step 1: Define variables for the shares. Let the share of the first born be F. Let the share of the second born be S. Let the share of the youngest born be Y. Step 2: Express S and Y in terms of F. S = (1)/(3)F Y = (1)/(6)F Step 3: Set up an equation for the total sum. The total sum is K1050, so: F + S + Y = 1050 Substitute the expressions for S and Y: F + (1)/(3)F + (1)/(6)F = 1050 Step 4: Solve for F. Find a common denominator for the fractions, which is 6: (6)/(6)F + (2)/(6)F + (1)/(6)F = 1050 Combine the terms: (6+2+1)/(6)F = 1050 (9)/(6)F = 1050 Simplify the fraction (9)/(6) to (3)/(2): (3)/(2)F = 1050 Multiply both sides by (2)/(3) to find F: F = 1050 × (2)/(3) F = (2100)/(3) F = 700 The first born's share is K700. Step 5: Calculate the shares of the second and youngest born. Second born's share (S): S = (1)/(3)F = (1)/(3) × 700 = (700)/(3) ≈ K233.33 Youngest born's share (Y): Y = (1)/(6)F = (1)/(6) × 700 = (700)/(6) = (350)/(3) ≈ K116.67 (Note: Shares are rounded to two decimal places for currency.) Step 6: Verify the total sum. K700.00 + K233.33 + K116.67 = K1050.00 The shares are: First born: K700.00 Second born: K233.33 Youngest born: K116.67 3 done, 2 left today. You're making progress.