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

Mathematics

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6 parts
i
1
ii
24
iii
15
iv

2:1\text{2:1}

v
60
vi

374 kg, 561 kg, and 935 kg374 \text{ kg, } 561 \text{ kg, and } 935 \text{ kg}

Step-by-step solution

22 steps

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.

Ratio=Mass of stone:Mass of dog\text{Ratio} = \text{Mass of stone} : \text{Mass of dog} Ratio=30 kg:90 kg\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.

3030:9030\frac{30}{30} : \frac{90}{30} 1:31 : 3

The ratio is 1:3\boxed{\text{1:3}}.

s)

Step 1: Find the total number of parts in the ratio.

4+5=9 parts4 + 5 = 9 \text{ parts}

Step 2: Find the value of one part.

7209 parts=80 per part\frac{720}{9 \text{ parts}} = 80 \text{ per part}

Step 3: Calculate each share.
First share: 4×80=3204 \times 80 = 320
Second share: 5×80=4005 \times 80 = 400
The shares are 320 and 400\boxed{320 \text{ and } 400}.

Part i· 5 steps

  1. Convert the mixed number to an improper fraction.

    112=321 \frac{1}{2} = \frac{3}{2}

  2. Write the proportion.

    q:32=20:30q : \frac{3}{2} = 20 : 30

  3. Express the proportion as a fraction.

    q32=2030\frac{q}{\frac{3}{2}} = \frac{20}{30}

  4. Simplify the right side of the equation.

    q32=23\frac{q}{\frac{3}{2}} = \frac{2}{3}

  5. Multiply both sides by 32\frac{3}{2} to solve for q.

    q=23×32q = \frac{2}{3} \times \frac{3}{2} q=1q = 1

    The value of q is 1\boxed{1}.

Part ii24

  1. Write the proportion.

    34:12=W:16\frac{3}{4} : \frac{1}{2} = W : 16

  2. Express the proportion as a fraction.

    3412=W16\frac{\frac{3}{4}}{\frac{1}{2}} = \frac{W}{16}

  3. Simplify the left side of the equation.

    34÷12=34×21=64=32\frac{3}{4} \div \frac{1}{2} = \frac{3}{4} \times \frac{2}{1} = \frac{6}{4} = \frac{3}{2}

    So, the equation becomes:

    32=W16\frac{3}{2} = \frac{W}{16}

  4. Multiply both sides by 16 to solve for W.

    W=32×16W = \frac{3}{2} \times 16 W=3×8W = 3 \times 8 W=24W = 24

    The value of W is 24\boxed{24}.

Part iii15

  1. Write the proportion as a fraction.

    aa+3=56\frac{a}{a+3} = \frac{5}{6}

  2. Cross-multiply.

    6×a=5×(a+3)6 \times a = 5 \times (a+3) 6a=5a+156a = 5a + 15

  3. Subtract $5a$ from both sides.

    6a−5a=156a - 5a = 15 a=15a = 15

    The value of a is 15\boxed{15}.

Part iv

2:1\text{2:1}

  1. Write the expenditure of Asha and Mbaga.

    Asha's expenditure = sh. 24000
    Mbaga's expenditure = sh. 12000

  2. Form the ratio of their expenditure (Asha

    Mbaga).

    24000:1200024000 : 12000

  3. Simplify the ratio by dividing both sides by their greatest common divisor, which is 12000.

    2400012000:1200012000\frac{24000}{12000} : \frac{12000}{12000} 2:12 : 1

    The ratio of their expenditure is 2:1\boxed{\text{2:1}}.

Part v60

  1. Convert the decimal and fraction to simpler forms.

    0.5=120.5 = \frac{1}{2} 1530=12\frac{15}{30} = \frac{1}{2}

  2. Write the proportion with the simplified terms.

    12:12=60:B\frac{1}{2} : \frac{1}{2} = 60 : B

  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:B1 : 1 = 60 : B

  4. This means B must be equal to 60.

    B=60B = 60

    The value of B is 60\boxed{60}.

Part vi

374 kg, 561 kg, and 935 kg374 \text{ kg, } 561 \text{ kg, and } 935 \text{ kg}

  1. Find the total number of parts in the ratio.

    2+3+5=10 parts2 + 3 + 5 = 10 \text{ parts}

  2. Find the value of one part.

    1870 kg10 parts=187 kg per part\frac{1870 \text{ kg}}{10 \text{ parts}} = 187 \text{ kg per part}

  3. Calculate each share.

    First share: 2×187 kg=374 kg2 \times 187 \text{ kg} = 374 \text{ kg}
    Second share: 3×187 kg=561 kg3 \times 187 \text{ kg} = 561 \text{ kg}
    Third share: 5×187 kg=935 kg5 \times 187 \text{ kg} = 935 \text{ kg}
    The shares are 374 kg, 561 kg, and 935 kg\boxed{374 \text{ kg, } 561 \text{ kg, and } 935 \text{ kg}}.

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