Find a common denominator for the fractions.

Question

Find a common denominator for the fractions.

Asked on March 28, 2026Mathematics

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.

Accepted Answer · 2026-03-28T08:34:51.685Z

4. a) Step 1: Find a common denominator for the fractions. The denominators are 5 and 7. The least common multiple of 5 and 7 is $5 \times 7 = 35$. Step 2: Convert the fractions to have the common denominator. $$\frac{3}{5} = \frac{3 \times 7}{5 \times 7} = \frac{21}{35}$$ $$\frac{12}{7} = \frac{12 \times 5}{7 \times 5} = \frac{60}{35}$$ Step 3: Add the fractions. $$\frac{21}{35} + \frac{60}{35} = \frac{21 + 60}{35} = \frac{81}{35}$$ The fraction is in its simplest form as 81 and 35 have no common factors other than 1. It can also be expressed as a mixed number. $$81 \div 35 = 2 \text{ with a remainder of } 11$$ So, $\frac{81}{35} = 2\frac{11}{35}$. The simplified expression is $\boxed{\frac{81}{35}}$ or $\boxed{2\frac{11}{35}}$. 4. b) Step 1: Determine the cost per kg of meat. Given that 3 kg of meat costs Gh¢60. Cost per kg = $\frac{\text{Total Cost}}{\text{Weight}} = \frac{60 \text{ Gh¢}}{3 \text{ kg}} = 20 \text{ Gh¢/kg}$. Step 2: Calculate the cost for each weight in the table. For 2 kg: Cost = $2 \text{ kg} \times 20 \text{ Gh¢/kg} = 40 \text{ Gh¢}$. For 5 kg: Cost = $5 \text{ kg} \times 20 \text{ Gh¢/kg} = 100 \text{ Gh¢}$. For 12 kg: Cost = $12 \text{ kg} \times 20 \text{ Gh¢/kg} = 240 \text{ Gh¢}$. Step 3: Complete the table. | Meat (kg) | 2 | 3 | 5 | 12 | |---|---|---|---|---| | Cost Gh¢ | 40 | 60 | 100 | 240 | 4. c) Step 1: Substitute each given value of $x$ into the relation $y = 3x - 6$ to find the corresponding $y$ value. For $x = -2$: $$y = 3(-2) - 6 = -6 - 6 = -12$$ For $x = -1$: $$y = 3(-1) - 6 = -3 - 6 = -9$$ For $x = 0$: $$y = 3(0) - 6 = 0 - 6 = -6$$ For $x = 1$: $$y = 3(1) - 6 = 3 - 6 = -3$$ For $x = 2$: $$y = 3(2) - 6 = 6 - 6 = 0$$ Step 2: Complete the table. | x | -2 | -1 | 0 | 1 | 2 | |---|---|---|---|---|---| | y | -12 | -9 | -6 | -3 | 0 | 5. a) Step 1: Identify the length and width of the rectangle. Length $L = 12.8$ m Width $W = 7.5$ m Step 2: Use the formula for the perimeter of a rectangle, $P = 2(L + W)$. $$P = 2(12.8 \text{ m} + 7.5 \text{ m})$$ $$P = 2(20.3 \text{ m})$$ $$P = 40.6 \text{ m}$$ The perimeter of the rectangle is $\boxed{40.6 \text{ m}}$. 5. b) Step 1: Simplify the expression $-30abc \div 60b^3c^2$. $$\frac{-30abc}{60b^3c^2}$$ Divide the numerical coefficients: $$\frac{-30}{60} = -\frac{1}{2}$$ Divide the variable terms using the rule $x^m/x^n = x^{m-n}$: $$a^1/1 = a$$ $$b^1/b^3 = b^{1-3} = b^{-2} = \frac{1}{b^2}$$ $$c^1/c^2 = c^{1-2} = c^{-1} = \frac{1}{c}$$ Combine the simplified terms: $$-\frac{1}{2} \cdot a \cdot \frac{1}{b^2} \cdot \frac{1}{c} = -\frac{a}{2b^2c}$$ Step 2: Substitute the given values $a=1$, $b=-2$, and $c=3$ into the simplified expression. $$-\frac{1}{2(-2)^2(3)}$$ $$-\frac{1}{2(4)(3)}$$ $$-\frac{1}{8(3)}$$ $$-\frac{1}{24}$$ The simplified and evaluated expression is $\boxed{-\frac{1}{24}}$. 5. c) Step 1: Understand that supplementary angles add up to $180^\circ$. The two angles given are $(3y + 5)^\circ$ and $(2y + 15)^\circ$. Step 2: Set up the equation. $$(3y + 5) + (2y + 15) = 180$$ Step 3: Combine like terms. $$5y + 20 = 180$$ Step 4: Isolate the term with $y$. $$5y = 180 - 20$$ $$5y = 160$$ Step 5: Solve for $y$. $$y = \frac{160}{5}$$ $$y = 32$$ The value of $y$ is $\boxed{32}$.

Accepted Answer · 2026-03-28T08:34:51.685Z

4. a) Step 1: Find a common denominator for the fractions. The denominators are 5 and 7. The least common multiple of 5 and 7 is $5 \times 7 = 35$. Step 2: Convert the fractions to have the common denominator. $$\frac{3}{5} = \frac{3 \times 7}{5 \times 7} = \frac{21}{35}$$ $$\frac{12}{7} = \frac{12 \times 5}{7 \times 5} = \frac{60}{35}$$ Step 3: Add the fractions. $$\frac{21}{35} + \frac{60}{35} = \frac{21 + 60}{35} = \frac{81}{35}$$ The fraction is in its simplest form as 81 and 35 have no common factors other than 1. It can also be expressed as a mixed number. $$81 \div 35 = 2 \text{ with a remainder of } 11$$ So, $\frac{81}{35} = 2\frac{11}{35}$. The simplified expression is $\boxed{\frac{81}{35}}$ or $\boxed{2\frac{11}{35}}$. 4. b) Step 1: Determine the cost per kg of meat. Given that 3 kg of meat costs Gh¢60. Cost per kg = $\frac{\text{Total Cost}}{\text{Weight}} = \frac{60 \text{ Gh¢}}{3 \text{ kg}} = 20 \text{ Gh¢/kg}$. Step 2: Calculate the cost for each weight in the table. For 2 kg: Cost = $2 \text{ kg} \times 20 \text{ Gh¢/kg} = 40 \text{ Gh¢}$. For 5 kg: Cost = $5 \text{ kg} \times 20 \text{ Gh¢/kg} = 100 \text{ Gh¢}$. For 12 kg: Cost = $12 \text{ kg} \times 20 \text{ Gh¢/kg} = 240 \text{ Gh¢}$. Step 3: Complete the table. | Meat (kg) | 2 | 3 | 5 | 12 | |---|---|---|---|---| | Cost Gh¢ | 40 | 60 | 100 | 240 | 4. c) Step 1: Substitute each given value of $x$ into the relation $y = 3x - 6$ to find the corresponding $y$ value. For $x = -2$: $$y = 3(-2) - 6 = -6 - 6 = -12$$ For $x = -1$: $$y = 3(-1) - 6 = -3 - 6 = -9$$ For $x = 0$: $$y = 3(0) - 6 = 0 - 6 = -6$$ For $x = 1$: $$y = 3(1) - 6 = 3 - 6 = -3$$ For $x = 2$: $$y = 3(2) - 6 = 6 - 6 = 0$$ Step 2: Complete the table. | x | -2 | -1 | 0 | 1 | 2 | |---|---|---|---|---|---| | y | -12 | -9 | -6 | -3 | 0 | 5. a) Step 1: Identify the length and width of the rectangle. Length $L = 12.8$ m Width $W = 7.5$ m Step 2: Use the formula for the perimeter of a rectangle, $P = 2(L + W)$. $$P = 2(12.8 \text{ m} + 7.5 \text{ m})$$ $$P = 2(20.3 \text{ m})$$ $$P = 40.6 \text{ m}$$ The perimeter of the rectangle is $\boxed{40.6 \text{ m}}$. 5. b) Step 1: Simplify the expression $-30abc \div 60b^3c^2$. $$\frac{-30abc}{60b^3c^2}$$ Divide the numerical coefficients: $$\frac{-30}{60} = -\frac{1}{2}$$ Divide the variable terms using the rule $x^m/x^n = x^{m-n}$: $$a^1/1 = a$$ $$b^1/b^3 = b^{1-3} = b^{-2} = \frac{1}{b^2}$$ $$c^1/c^2 = c^{1-2} = c^{-1} = \frac{1}{c}$$ Combine the simplified terms: $$-\frac{1}{2} \cdot a \cdot \frac{1}{b^2} \cdot \frac{1}{c} = -\frac{a}{2b^2c}$$ Step 2: Substitute the given values $a=1$, $b=-2$, and $c=3$ into the simplified expression. $$-\frac{1}{2(-2)^2(3)}$$ $$-\frac{1}{2(4)(3)}$$ $$-\frac{1}{8(3)}$$ $$-\frac{1}{24}$$ The simplified and evaluated expression is $\boxed{-\frac{1}{24}}$. 5. c) Step 1: Understand that supplementary angles add up to $180^\circ$. The two angles given are $(3y + 5)^\circ$ and $(2y + 15)^\circ$. Step 2: Set up the equation. $$(3y + 5) + (2y + 15) = 180$$ Step 3: Combine like terms. $$5y + 20 = 180$$ Step 4: Isolate the term with $y$. $$5y = 180 - 20$$ $$5y = 160$$ Step 5: Solve for $y$. $$y = \frac{160}{5}$$ $$y = 32$$ The value of $y$ is $\boxed{32}$.