Here are the solutions to the problems. 5. Evaluate $\frac{3}{\sqrt{0.0136}} - (3.72)^2$ correct to 4 s.f. Note: As I do not have access to physical tables, I will perform the calculations directly and round the final answer to 4 significant figures. Step 1: Evaluate $\sqrt{0.0136}$. $$ \sqrt{0.0136} \approx 0.11662076 $$ Step 2: Evaluate $\frac{3}{\sqrt{0.0136}}$. $$ \frac{3}{0.11662076} \approx 25.72457 $$ Step 3: Evaluate $(3.72)^2$. $$ (3.72)^2 = 13.8384 $$ Step 4: Perform the subtraction. $$ 25.72457 - 13.8384 = 11.88617 $$ Step 5: Round the result to 4 significant figures. $$ 11.88617 \approx 11.89 $$ The value is $\boxed{11.89}$. 6. Simplify as far as possible $\frac{3}{x-y} - \frac{1}{x+y}$ Step 1: Find a common denominator, which is $(x-y)(x+y)$. Step 2: Rewrite each fraction with the common denominator. $$ \frac{3}{x-y} - \frac{1}{x+y} = \frac{3(x+y)}{(x-y)(x+y)} - \frac{1(x-y)}{(x+y)(x-y)} $$ Step 3: Combine the numerators over the common denominator. $$ \frac{3(x+y) - 1(x-y)}{(x-y)(x+y)} $$ Step 4: Expand the numerator and the denominator. Numerator: $3x + 3y - x + y = 2x + 4y$ Denominator: $(x-y)(x+y) = x^2 - y^2$ Step 5: Write the simplified expression. $$ \frac{2x+4y}{x^2-y^2} $$ The simplified expression is $\boxed{\frac{2x+4y}{x^2-y^2}}$. 7. Fifteen men working for eight hours a day can complete a certain job in exactly 24 days. For how many hours a day must sixteen men work in order to complete the same job in exactly 20 days. This is an inverse proportion problem. The total work done is constant. Let $M$ be the number of men, $H$ be the hours per day, and $D$ be the number of days. The total work is proportional to $M \times H \times D$. Step 1: Set up the equation based on the given information. $$ M_1 \times H_1 \times D_1 = M_2 \times H_2 \times D_2 $$ Given: $M_1 = 15$ men $H_1 = 8$ hours/day $D_1 = 24$ days $M_2 = 16$ men $H_2 = ?$ hours/day $D_2 = 20$ days Step 2: Substitute the known values into the equation. $$ 15 \times 8 \times 24 = 16 \times H_2 \times 20 $$ Step 3: Calculate the left side of the equation. $$ 15 \times 8 \times 24 = 120 \times 24 = 2880 $$ Step 4: Simplify the equation and solve for $H_2$. $$ 2880 = 320 \times H_2 $$ $$ H_2 = \frac{2880}{320} $$ $$ H_2 = 9 $$ Sixteen men must work 9 hours a day. The number of hours per day is $\boxed{9 \text{ hours}}$. That's 2 down. 3 left today — send the next one.