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Sagot :
Sure! Let's work through the problem step by step.
### Part (a)
Jamie has a 28-meter long rope and cuts it into two parts with a ratio of 4:3.
1. Determine the total number of parts:
The ratio 4:3 means that the rope is divided into [tex]\( 4 + 3 = 7 \)[/tex] parts in total.
2. Calculate the length of each part:
- First part (ratio 4):
[tex]\[ \text{First part length} = \frac{4}{7} \times 28 = \frac{4 \times 28}{7} = 16 \text{ meters} \][/tex]
- Second part (ratio 3):
[tex]\[ \text{Second part length} = \frac{3}{7} \times 28 = \frac{3 \times 28}{7} = 12 \text{ meters} \][/tex]
So, the lengths of the two parts are 16 meters and 12 meters respectively.
### Part (b)
Jamie now cuts a piece of the same length from each part of these ropes. After this cut, the remaining lengths have a ratio of 5:3. We need to find the length of the piece that Jamie cuts from each part.
1. Initial lengths:
[tex]\[ \text{First part} = 16 \text{ meters}, \text{ Second part} = 12 \text{ meters} \][/tex]
2. Let [tex]\( x \)[/tex] be the length of the piece cut from each part. After cutting, the lengths of the parts will be:
[tex]\[ \text{First part remaining} = 16 - x \text{ meters} \][/tex]
[tex]\[ \text{Second part remaining} = 12 - x \text{ meters} \][/tex]
3. The ratio of the remaining lengths is given as 5:3, so we set up the following equation:
[tex]\[ \frac{16 - x}{12 - x} = \frac{5}{3} \][/tex]
4. Solve the equation for [tex]\( x \)[/tex]:
- Cross multiply to get:
[tex]\[ 3(16 - x) = 5(12 - x) \][/tex]
- Distribute and simplify:
[tex]\[ 48 - 3x = 60 - 5x \][/tex]
- Combine like terms:
[tex]\[ 2x = 12 \][/tex]
- Solve for [tex]\( x \)[/tex]:
[tex]\[ x = 6 \][/tex]
Thus, the length of the piece that Jamie cuts from each part of the rope is 6 meters.
5. Verify the remaining lengths:
- Remaining length of the first part:
[tex]\[ 16 - 6 = 10 \text{ meters} \][/tex]
- Remaining length of the second part:
[tex]\[ 12 - 6 = 6 \text{ meters} \][/tex]
The ratio of the remaining lengths is:
[tex]\[ \frac{10}{6} = \frac{5}{3} \][/tex]
which confirms our calculation.
### Summary
(a) The lengths of the two parts of the rope are 16 meters and 12 meters.
(b) The length that Jamie cuts from each part of the rope is 6 meters.
### Part (a)
Jamie has a 28-meter long rope and cuts it into two parts with a ratio of 4:3.
1. Determine the total number of parts:
The ratio 4:3 means that the rope is divided into [tex]\( 4 + 3 = 7 \)[/tex] parts in total.
2. Calculate the length of each part:
- First part (ratio 4):
[tex]\[ \text{First part length} = \frac{4}{7} \times 28 = \frac{4 \times 28}{7} = 16 \text{ meters} \][/tex]
- Second part (ratio 3):
[tex]\[ \text{Second part length} = \frac{3}{7} \times 28 = \frac{3 \times 28}{7} = 12 \text{ meters} \][/tex]
So, the lengths of the two parts are 16 meters and 12 meters respectively.
### Part (b)
Jamie now cuts a piece of the same length from each part of these ropes. After this cut, the remaining lengths have a ratio of 5:3. We need to find the length of the piece that Jamie cuts from each part.
1. Initial lengths:
[tex]\[ \text{First part} = 16 \text{ meters}, \text{ Second part} = 12 \text{ meters} \][/tex]
2. Let [tex]\( x \)[/tex] be the length of the piece cut from each part. After cutting, the lengths of the parts will be:
[tex]\[ \text{First part remaining} = 16 - x \text{ meters} \][/tex]
[tex]\[ \text{Second part remaining} = 12 - x \text{ meters} \][/tex]
3. The ratio of the remaining lengths is given as 5:3, so we set up the following equation:
[tex]\[ \frac{16 - x}{12 - x} = \frac{5}{3} \][/tex]
4. Solve the equation for [tex]\( x \)[/tex]:
- Cross multiply to get:
[tex]\[ 3(16 - x) = 5(12 - x) \][/tex]
- Distribute and simplify:
[tex]\[ 48 - 3x = 60 - 5x \][/tex]
- Combine like terms:
[tex]\[ 2x = 12 \][/tex]
- Solve for [tex]\( x \)[/tex]:
[tex]\[ x = 6 \][/tex]
Thus, the length of the piece that Jamie cuts from each part of the rope is 6 meters.
5. Verify the remaining lengths:
- Remaining length of the first part:
[tex]\[ 16 - 6 = 10 \text{ meters} \][/tex]
- Remaining length of the second part:
[tex]\[ 12 - 6 = 6 \text{ meters} \][/tex]
The ratio of the remaining lengths is:
[tex]\[ \frac{10}{6} = \frac{5}{3} \][/tex]
which confirms our calculation.
### Summary
(a) The lengths of the two parts of the rope are 16 meters and 12 meters.
(b) The length that Jamie cuts from each part of the rope is 6 meters.
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