Class 7 Algebra-Expressions and Equations worksheet featuring multiple-choice and short-answer questions on algebraic concepts.

Let's solve each question one by one with detailed explanations.

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(1) What are the two numbers whose sum is 63 and their difference is 19?

Let the two numbers be $ x $ and $ y $, where $ x > y $.

We are given:
- $ x + y = 63 $ (Equation 1)
- $ x - y = 19 $ (Equation 2)

Add both equations:
$$
(x + y) + (x - y) = 63 + 19 \\
2x = 82 \Rightarrow x = 41
$$

Substitute $ x = 41 $ into Equation 1:
$$
41 + y = 63 \Rightarrow y = 22
$$

✔ Answer: The two numbers are 41 and 22.

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(2) Gautam is deciding to buy a car. He can choose between a diesel car and a petrol car. The diesel car costs Rs.9540 more than the petrol car. However, it can travel 17 km per litre, whereas the petrol car can only travel 11 km per litre. Also, diesel costs Rs.57.8 per litre, and petrol costs Rs.57.2 per litre. If he buys a diesel car, how far must he drive before the savings in fuel covers the extra price of the diesel car?

Let’s find out how much fuel cost per km for each car:

#### Diesel Car:
- Distance per litre: 17 km
- Cost per litre: ₹57.8
- So, cost per km = $ \frac{57.8}{17} = ₹3.394 $ per km

#### Petrol Car:
- Distance per litre: 11 km
- Cost per litre: ₹57.2
- So, cost per km = $ \frac{57.2}{11} = ₹5.2 $ per km

Now, savings per km if using diesel instead of petrol:
$$
5.2 - 3.394 = ₹1.806 \text{ per km}
$$

Extra cost of diesel car = ₹9540

So, distance needed to recover this cost:
$$
\frac{9540}{1.806} \approx 5282.3 \text{ km}
$$

Rounding to nearest whole number: 5282 km

✔ Answer: He must drive approximately 5282 km.

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(3) If $ x = 4 $ and $ y = x + 9 $, then what is the value of $ x^2 + y^2 $?

Given:
- $ x = 4 $
- $ y = x + 9 = 4 + 9 = 13 $

Now compute:
$$
x^2 + y^2 = 4^2 + 13^2 = 16 + 169 = 185
$$

✔ Answer: 185

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(4) A number if multiplied by 5 and 22 is added to the product. If result is 87, what was the original number?

Let the number be $ x $.

According to the problem:
$$
5x + 22 = 87
$$

Solve:
$$
5x = 87 - 22 = 65 \Rightarrow x = \frac{65}{5} = 13
$$

✔ Answer: The original number is 13

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(5) Sum of digits in a two-digit number is 9. If 27 is subtracted from the number, digits of the number are interchanged. Find the original number.

Let the two-digit number have tens digit $ x $ and units digit $ y $. Then:

- Number = $ 10x + y $
- Sum of digits: $ x + y = 9 $ → Equation (1)
- After subtracting 27, digits are reversed: $ 10x + y - 27 = 10y + x $ → Equation (2)

Simplify Equation (2):
$$
10x + y - 27 = 10y + x \\
10x - x + y - 10y = 27 \\
9x - 9y = 27 \Rightarrow x - y = 3 \quad \text{(Divide by 9)}
$$

Now solve:
- $ x + y = 9 $
- $ x - y = 3 $

Add both:
$$
2x = 12 \Rightarrow x = 6 \Rightarrow y = 3
$$

So, number = $ 10x + y = 60 + 3 = 63 $

Check:
- Sum of digits: $ 6 + 3 = 9 $ ✔
- Subtract 27: $ 63 - 27 = 36 $, which is reverse of 63 ✔

✔ Answer: Original number is 63

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(6) If $ a + b = 13 $, $ b + c = 24 $, and $ c + a = 19 $, then what is the value of $ a + b + c $?

We are given:
1. $ a + b = 13 $
2. $ b + c = 24 $
3. $ c + a = 19 $

Add all three equations:
$$
(a + b) + (b + c) + (c + a) = 13 + 24 + 19 = 56 \\
2a + 2b + 2c = 56 \Rightarrow 2(a + b + c) = 56 \\
a + b + c = 28
$$

✔ Answer: 28

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(7) A man lost 12% of his salary and spent 8% of the remaining still has Rs.140. His total salary was?

Let total salary be $ x $

- Lost 12%, so remaining = $ 88\% $ of $ x = 0.88x $
- Spent 8% of remaining → so he spends $ 0.08 \times 0.88x = 0.0704x $
- Amount left = $ 0.88x - 0.0704x = 0.8096x $

But this equals ₹140:
$$
0.8096x = 140 \Rightarrow x = \frac{140}{0.8096} \approx 172.925
$$

So, salary ≈ ₹172.925 → which is ₹172.925, close to option c. Rs.172.925

✔ Answer: c. Rs.172.925

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(8) Gautam bought a total of 20 hens and ducks. He got a discount of Rs. 0.80 for each hen and a discount of Rs. 1.50 for each duck. He saved Rs. 23.70 in total. How many hens did he buy?

Let:
- Number of hens = $ h $
- Number of ducks = $ d $

Given:
- $ h + d = 20 $ → Equation (1)
- Total savings: $ 0.80h + 1.50d = 23.70 $ → Equation (2)

From (1): $ d = 20 - h $

Substitute into (2):
$$
0.80h + 1.50(20 - h) = 23.70 \\
0.80h + 30 - 1.50h = 23.70 \\
-0.70h = 23.70 - 30 = -6.30 \\
h = \frac{-6.30}{-0.70} = 9
$$

So, number of hens = 9

Check:
- Hens = 9 → savings = $ 9 \times 0.80 = 7.20 $
- Ducks = 11 → savings = $ 11 \times 1.50 = 16.50 $
- Total = $ 7.20 + 16.50 = 23.70 $ ✔

✔ Answer: b. 9

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✔ Final Answers Summary:

| Question | Answer |
|--------|--------|
| (1) | 41 and 22 |
| (2) | Approximately 5282 km |
| (3) | 185 |
| (4) | 13 |
| (5) | 63 |
| (6) | 28 |
| (7) | c. Rs.172.925 |
| (8) | b. 9 |

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