Class 8 Maths Direct and Inverse Proportion Worksheet - Free Printable

Problem 1: Which of the following are in Direct or Inverse proportion?

#### (a)
| $x$ | 4 | 5 | 6 |
|-----|---|---|---|
| $y$ | 10 | 12 | 14 |

- Check for Direct Proportion:
In direct proportion, the ratio $\frac{y}{x}$ should be constant.
- For $x = 4$, $y = 10$: $\frac{y}{x} = \frac{10}{4} = 2.5$
- For $x = 5$, $y = 12$: $\frac{y}{x} = \frac{12}{5} = 2.4$
- For $x = 6$, $y = 14$: $\frac{y}{x} = \frac{14}{6} \approx 2.33$

Since the ratio $\frac{y}{x}$ is not constant, $x$ and $y$ are not in direct proportion.

- Check for Inverse Proportion:
In inverse proportion, the product $xy$ should be constant.
- For $x = 4$, $y = 10$: $xy = 4 \times 10 = 40$
- For $x = 5$, $y = 12$: $xy = 5 \times 12 = 60$
- For $x = 6$, $y = 14$: $xy = 6 \times 14 = 84$

Since the product $xy$ is not constant, $x$ and $y$ are not in inverse proportion.

Conclusion: Neither direct nor inverse proportion.

#### (b)
| $x$ | 4 | 5 | 6 |
|-----|---|---|---|
| $y$ | 12 | 10 | 8 |

- Check for Direct Proportion:
- For $x = 4$, $y = 12$: $\frac{y}{x} = \frac{12}{4} = 3$
- For $x = 5$, $y = 10$: $\frac{y}{x} = \frac{10}{5} = 2$
- For $x = 6$, $y = 8$: $\frac{y}{x} = \frac{8}{6} \approx 1.33$

Since the ratio $\frac{y}{x}$ is not constant, $x$ and $y$ are not in direct proportion.

- Check for Inverse Proportion:
- For $x = 4$, $y = 12$: $xy = 4 \times 12 = 48$
- For $x = 5$, $y = 10$: $xy = 5 \times 10 = 50$
- For $x = 6$, $y = 8$: $xy = 6 \times 8 = 48$

The product $xy$ is approximately constant (with a slight discrepancy due to rounding), so $x$ and $y$ are in inverse proportion.

#### (c) Distance travelled and petrol consumed by a car
- Generally, the distance travelled is directly proportional to the amount of petrol consumed (assuming a constant fuel efficiency).
Conclusion: Direct proportion.

#### (d) Area of a piece of land and price of land
- The price of land is typically directly proportional to its area (assuming a constant price per unit area).
Conclusion: Direct proportion.

#### (e) Speed of a car and time taken to cover a fixed distance
- Speed and time are inversely proportional because if the speed increases, the time taken to cover a fixed distance decreases.
Conclusion: Inverse proportion.

Final Answer for Problem 1:
(a) Neither
(b) Inverse
(c) Direct
(d) Direct
(e) Inverse

$$
\boxed{\text{(a) Neither, (b) Inverse, (c) Direct, (d) Direct, (e) Inverse}}
$$

---

Problem 2: Find the value of $x$ and $y$ if $x : y = 2 : 3$ and $2 : x = 1 : 2$

#### Given:
1. $x : y = 2 : 3$
This implies $\frac{x}{y} = \frac{2}{3}$, or $x = \frac{2}{3}y$.

2. $2 : x = 1 : 2$
This implies $\frac{2}{x} = \frac{1}{2}$, or $x = 4$.

#### Solve for $y$:
Using $x = \frac{2}{3}y$ and substituting $x = 4$:
$$
4 = \frac{2}{3}y \implies y = 4 \times \frac{3}{2} = 6.
$$

Final Answer for Problem 2:
$$
\boxed{x = 4, y = 6}
$$

---

Problem 3: A car is moving at a uniform speed of 45 km/h.

#### (a) How far will it travel in 5 hours?
- Distance = Speed × Time
$$ \text{Distance} = 45 \, \text{km/h} \times 5 \, \text{h} = 225 \, \text{km}. $$

#### (b) Find the distance covered in 40 minutes.
- Convert 40 minutes to hours:
$$ 40 \, \text{minutes} = \frac{40}{60} \, \text{hours} = \frac{2}{3} \, \text{hours}. $$
- Distance = Speed × Time
$$ \text{Distance} = 45 \, \text{km/h} \times \frac{2}{3} \, \text{h} = 30 \, \text{km}. $$

Final Answers for Problem 3:
(a) $ \boxed{225} $ km
(b) $ \boxed{30} $ km

---

Problem 4: In a students' hostel with 12 students, a fixed quantity of food lasts for 45 days. If 16 more students join in, then the food will last for how many days?

#### Given:
- Initially, 12 students have food for 45 days.
- Total food available = $12 \times 45 = 540$ student-days (since each student consumes food for one day).

#### After 16 more students join:
- Total number of students = $12 + 16 = 28$.
- Let the food last for $d$ days.
Total food consumed = $28 \times d$ student-days.

Since the total food available is 540 student-days:
$$
28 \times d = 540 \implies d = \frac{540}{28} = \frac{270}{14} = \frac{135}{7} \approx 19.29 \, \text{days}.
$$

Final Answer for Problem 4:
$$
\boxed{19.29}
$$

---

Problem 5: A gardener can plant 30 plants in 2 hours. How many more gardeners should he appoint so that they can plant 150 plants along the street in 2 hours?

#### Given:
- One gardener plants 30 plants in 2 hours.
- Required: 150 plants in 2 hours.

#### Calculate the number of gardeners needed:
- Plants planted by one gardener in 2 hours = 30.
- Plants to be planted in 2 hours = 150.
- Number of gardeners required = $\frac{150}{30} = 5$.

#### Additional gardeners needed:
- Initially, there is 1 gardener.
- Additional gardeners = $5 - 1 = 4$.

Final Answer for Problem 5:
$$
\boxed{4}
$$

---

Problem 6: If $2 : 3 = x : 31$, find $x$.

#### Given:
$$
\frac{2}{3} = \frac{x}{31}.
$$

#### Solve for $x$:
Cross-multiply:
$$
2 \times 31 = 3 \times x \implies 62 = 3x \implies x = \frac{62}{3}.
$$

Final Answer for Problem 6:
$$
\boxed{\frac{62}{3}}
$$

---

Problem 7: If $x$ and $y$ are in inverse proportion, find the value of $a$, $b$, and $c$ in the table.

| $x$ | 25 | 15 | $b$ | 10 |
|-----|----|----|-----|----|
| $y$ | 3 | $a$ | 4 | $c$ |

#### Given:
- $x$ and $y$ are in inverse proportion, so $xy = k$ (constant).

#### Step 1: Find the constant $k$.
Using $x = 25$ and $y = 3$:
$$
k = x \cdot y = 25 \cdot 3 = 75.
$$

#### Step 2: Find $a$.
Using $x = 15$ and $xy = 75$:
$$
15 \cdot a = 75 \implies a = \frac{75}{15} = 5.
$$

#### Step 3: Find $b$.
Using $y = 4$ and $xy = 75$:
$$
b \cdot 4 = 75 \implies b = \frac{75}{4} = 18.75.
$$

#### Step 4: Find $c$.
Using $x = 10$ and $xy = 75$:
$$
10 \cdot c = 75 \implies c = \frac{75}{10} = 7.5.
$$

Final Answers for Problem 7:
$$
\boxed{a = 5, b = 18.75, c = 7.5}
$$

---

Problem 8: If $A : B = 2 : 3$ and $B : C = 4 : 5$, then find $A : C$.

#### Given:
1. $A : B = 2 : 3$
This implies $\frac{A}{B} = \frac{2}{3}$, or $A = \frac{2}{3}B$.

2. $B : C = 4 : 5$
This implies $\frac{B}{C} = \frac{4}{5}$, or $B = \frac{4}{5}C$.

#### Express $A$ in terms of $C$:
From $B = \frac{4}{5}C$, substitute into $A = \frac{2}{3}B$:
$$
A = \frac{2}{3} \left(\frac{4}{5}C\right) = \frac{8}{15}C.
$$

#### Ratio $A : C$:
$$
\frac{A}{C} = \frac{8}{15} \implies A : C = 8 : 15.
$$

Final Answer for Problem 8:
$$
\boxed{8 : 15}
$$

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Final Summary of All Answers:

1. (a) Neither, (b) Inverse, (c) Direct, (d) Direct, (e) Inverse
2. $x = 4$, $y = 6$
3. (a) 225 km, (b) 30 km
4. 19.29 days
5. 4
6. $\frac{62}{3}$
7. $a = 5$, $b = 18.75$, $c = 7.5$
8. $8 : 15$