Let's solve each problem step by step using
proportion and
inverse proportion concepts. Inverse proportion means that as one quantity increases, the other decreases (and vice versa), especially when dealing with work, time, and people.
---
Problem 1:
> An army camp has provisions for 240 men for 28 days. How long will the provisions last if only 112 men are sent to the camp?
#### Step-by-step:
- The total amount of provisions is fixed.
- More men → provisions last fewer days (inverse proportion).
- Let’s find how many days the provisions will last for 112 men.
We use the concept:
$$
\text{Men} \times \text{Days} = \text{Constant (total provisions)}
$$
So,
$$
240 \times 28 = 112 \times x
$$
Solve for $x$:
$$
x = \frac{240 \times 28}{112}
$$
Calculate:
- $240 \div 112 = 2.142...$, but better to simplify:
- $240 / 112 = 15/7$ (divide numerator and denominator by 16)
- So: $x = (15/7) \times 28 = 15 \times 4 = 60$
✔ Answer: 60 days
---
Problem 2:
> If 9 men can do a piece of work in 6 days, how long will it take 12 men to do the same work?
This is inverse proportion: more men → less time.
Use:
$$
\text{Men} \times \text{Days} = \text{Constant (work)}
$$
So:
$$
9 \times 6 = 12 \times x
$$
$$
x = \frac{54}{12} = 4.5
$$
✔ Answer: 4.5 days (or 4 days and 12 hours)
---
Problem 3:
> In a home economics class, 5 girls worked together and sewed a dress in 7 days. How long would it have taken 3 girls to make the same dress?
Again, inverse proportion: fewer girls → more time.
$$
5 \times 7 = 3 \times x
$$
$$
x = \frac{35}{3} \approx 11.67 \text{ days}
$$
✔ Answer: $ \frac{35}{3} $ days or approximately 11 days and 16 hours
---
Problem 4:
> If 5 boys can decorate a hall for a program in $3\frac{1}{2}$ hours, how long will it take 7 boys to decorate it?
First, convert $3\frac{1}{2}$ to improper fraction: $ \frac{7}{2} $ hours.
Now use:
$$
5 \times \frac{7}{2} = 7 \times x
$$
$$
x = \frac{5 \times 7}{2 \times 7} = \frac{35}{14} = 2.5 \text{ hours}
$$
Or $2.5 = 2\frac{1}{2}$ hours.
✔ Answer: $2\frac{1}{2}$ hours
---
Problem 5:
> A contractor hired 150 men to pave a road in 30 days. How many men would he have to hire to do the same work in 20 days?
Inverse proportion: fewer days → more men needed.
$$
150 \times 30 = x \times 20
$$
$$
x = \frac{150 \times 30}{20} = \frac{4500}{20} = 225
$$
✔ Answer: 225 men
---
✔ Final Answers:
1.
60 days
2.
4.5 days
3.
$ \frac{35}{3} $ days ≈ 11.67 days
4.
$ 2\frac{1}{2} $ hours
5.
225 men
Let me know if you'd like these explained with diagrams or tables!
Parent Tip: Review the logic above to help your child master the concept of inverse proportion worksheet.
