Write the ratio of the mass of the stone to the mass of the dog.

Question

Write the ratio of the mass of the stone to the mass of the dog.

Asked on March 29, 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 · 2026-03-29T08:57:58.917Z

Here are the solutions to the problems: r) Step 1: Write the ratio of the mass of the stone to the mass of the dog. $$ \text{Ratio} = \text{Mass of stone} : \text{Mass of dog} $$ $$ \text{Ratio} = 30 \text{ kg} : 90 \text{ kg} $$ Step 2: Simplify the ratio by dividing both sides by their greatest common divisor, which is 30. $$ \frac{30}{30} : \frac{90}{30} $$ $$ 1 : 3 $$ The ratio is $\boxed{\text{1:3}}$. s) Step 1: Find the total number of parts in the ratio. $$ 4 + 5 = 9 \text{ parts} $$ Step 2: Find the value of one part. $$ \frac{720}{9 \text{ parts}} = 80 \text{ per part} $$ Step 3: Calculate each share. First share: $4 \times 80 = 320$ Second share: $5 \times 80 = 400$ The shares are $\boxed{320 \text{ and } 400}$. Problem 2 (i) Step 1: Convert the mixed number to an improper fraction. $$ 1 \frac{1}{2} = \frac{3}{2} $$ Step 2: Write the proportion. $$ q : \frac{3}{2} = 20 : 30 $$ Step 3: Express the proportion as a fraction. $$ \frac{q}{\frac{3}{2}} = \frac{20}{30} $$ Step 4: Simplify the right side of the equation. $$ \frac{q}{\frac{3}{2}} = \frac{2}{3} $$ Step 5: Multiply both sides by $\frac{3}{2}$ to solve for q. $$ q = \frac{2}{3} \times \frac{3}{2} $$ $$ q = 1 $$ The value of q is $\boxed{1}$. (ii) Step 1: Write the proportion. $$ \frac{3}{4} : \frac{1}{2} = W : 16 $$ Step 2: Express the proportion as a fraction. $$ \frac{\frac{3}{4}}{\frac{1}{2}} = \frac{W}{16} $$ Step 3: Simplify the left side of the equation. $$ \frac{3}{4} \div \frac{1}{2} = \frac{3}{4} \times \frac{2}{1} = \frac{6}{4} = \frac{3}{2} $$ So, the equation becomes: $$ \frac{3}{2} = \frac{W}{16} $$ Step 4: Multiply both sides by 16 to solve for W. $$ W = \frac{3}{2} \times 16 $$ $$ W = 3 \times 8 $$ $$ W = 24 $$ The value of W is $\boxed{24}$. (iii) Step 1: Write the proportion as a fraction. $$ \frac{a}{a+3} = \frac{5}{6} $$ Step 2: Cross-multiply. $$ 6 \times a = 5 \times (a+3) $$ $$ 6a = 5a + 15 $$ Step 3: Subtract $5a$ from both sides. $$ 6a - 5a = 15 $$ $$ a = 15 $$ The value of a is $\boxed{15}$. (iv) Step 1: Write the expenditure of Asha and Mbaga. Asha's expenditure = sh. 24000 Mbaga's expenditure = sh. 12000 Step 2: Form the ratio of their expenditure (Asha : Mbaga). $$ 24000 : 12000 $$ Step 3: Simplify the ratio by dividing both sides by their greatest common divisor, which is 12000. $$ \frac{24000}{12000} : \frac{12000}{12000} $$ $$ 2 : 1 $$ The ratio of their expenditure is $\boxed{\text{2:1}}$. (v) Step 1: Convert the decimal and fraction to simpler forms. $$ 0.5 = \frac{1}{2} $$ $$ \frac{15}{30} = \frac{1}{2} $$ Step 2: Write the proportion with the simplified terms. $$ \frac{1}{2} : \frac{1}{2} = 60 : B $$ Step 3: Since the ratio on the left side is $1:1$, the ratio on the right side must also be $1:1$. $$ 1 : 1 = 60 : B $$ Step 4: This means B must be equal to 60. $$ B = 60 $$ The value of B is $\boxed{60}$. (vi) Step 1: Find the total number of parts in the ratio. $$ 2 + 3 + 5 = 10 \text{ parts} $$ Step 2: Find the value of one part. $$ \frac{1870 \text{ kg}}{10 \text{ parts}} = 187 \text{ kg per part} $$ Step 3: Calculate each share. First share: $2 \times 187 \text{ kg} = 374 \text{ kg}$ Second share: $3 \times 187 \text{ kg} = 561 \text{ kg}$ Third share: $5 \times 187 \text{ kg} = 935 \text{ kg}$ The shares are $\boxed{374 \text{ kg, } 561 \text{ kg, and } 935 \text{ kg}}$.

Accepted Answer · 2026-03-29T08:57:58.917Z

Here are the solutions to the problems: r) Step 1: Write the ratio of the mass of the stone to the mass of the dog. $$ \text{Ratio} = \text{Mass of stone} : \text{Mass of dog} $$ $$ \text{Ratio} = 30 \text{ kg} : 90 \text{ kg} $$ Step 2: Simplify the ratio by dividing both sides by their greatest common divisor, which is 30. $$ \frac{30}{30} : \frac{90}{30} $$ $$ 1 : 3 $$ The ratio is $\boxed{\text{1:3}}$. s) Step 1: Find the total number of parts in the ratio. $$ 4 + 5 = 9 \text{ parts} $$ Step 2: Find the value of one part. $$ \frac{720}{9 \text{ parts}} = 80 \text{ per part} $$ Step 3: Calculate each share. First share: $4 \times 80 = 320$ Second share: $5 \times 80 = 400$ The shares are $\boxed{320 \text{ and } 400}$. Problem 2 (i) Step 1: Convert the mixed number to an improper fraction. $$ 1 \frac{1}{2} = \frac{3}{2} $$ Step 2: Write the proportion. $$ q : \frac{3}{2} = 20 : 30 $$ Step 3: Express the proportion as a fraction. $$ \frac{q}{\frac{3}{2}} = \frac{20}{30} $$ Step 4: Simplify the right side of the equation. $$ \frac{q}{\frac{3}{2}} = \frac{2}{3} $$ Step 5: Multiply both sides by $\frac{3}{2}$ to solve for q. $$ q = \frac{2}{3} \times \frac{3}{2} $$ $$ q = 1 $$ The value of q is $\boxed{1}$. (ii) Step 1: Write the proportion. $$ \frac{3}{4} : \frac{1}{2} = W : 16 $$ Step 2: Express the proportion as a fraction. $$ \frac{\frac{3}{4}}{\frac{1}{2}} = \frac{W}{16} $$ Step 3: Simplify the left side of the equation. $$ \frac{3}{4} \div \frac{1}{2} = \frac{3}{4} \times \frac{2}{1} = \frac{6}{4} = \frac{3}{2} $$ So, the equation becomes: $$ \frac{3}{2} = \frac{W}{16} $$ Step 4: Multiply both sides by 16 to solve for W. $$ W = \frac{3}{2} \times 16 $$ $$ W = 3 \times 8 $$ $$ W = 24 $$ The value of W is $\boxed{24}$. (iii) Step 1: Write the proportion as a fraction. $$ \frac{a}{a+3} = \frac{5}{6} $$ Step 2: Cross-multiply. $$ 6 \times a = 5 \times (a+3) $$ $$ 6a = 5a + 15 $$ Step 3: Subtract $5a$ from both sides. $$ 6a - 5a = 15 $$ $$ a = 15 $$ The value of a is $\boxed{15}$. (iv) Step 1: Write the expenditure of Asha and Mbaga. Asha's expenditure = sh. 24000 Mbaga's expenditure = sh. 12000 Step 2: Form the ratio of their expenditure (Asha : Mbaga). $$ 24000 : 12000 $$ Step 3: Simplify the ratio by dividing both sides by their greatest common divisor, which is 12000. $$ \frac{24000}{12000} : \frac{12000}{12000} $$ $$ 2 : 1 $$ The ratio of their expenditure is $\boxed{\text{2:1}}$. (v) Step 1: Convert the decimal and fraction to simpler forms. $$ 0.5 = \frac{1}{2} $$ $$ \frac{15}{30} = \frac{1}{2} $$ Step 2: Write the proportion with the simplified terms. $$ \frac{1}{2} : \frac{1}{2} = 60 : B $$ Step 3: Since the ratio on the left side is $1:1$, the ratio on the right side must also be $1:1$. $$ 1 : 1 = 60 : B $$ Step 4: This means B must be equal to 60. $$ B = 60 $$ The value of B is $\boxed{60}$. (vi) Step 1: Find the total number of parts in the ratio. $$ 2 + 3 + 5 = 10 \text{ parts} $$ Step 2: Find the value of one part. $$ \frac{1870 \text{ kg}}{10 \text{ parts}} = 187 \text{ kg per part} $$ Step 3: Calculate each share. First share: $2 \times 187 \text{ kg} = 374 \text{ kg}$ Second share: $3 \times 187 \text{ kg} = 561 \text{ kg}$ Third share: $5 \times 187 \text{ kg} = 935 \text{ kg}$ The shares are $\boxed{374 \text{ kg, } 561 \text{ kg, and } 935 \text{ kg}}$.