Grade 7 - Algebra-Expressions and Equations | Math Practice ... - Free Printable

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 when using diesel instead of petrol:
$$
₹5.2 - ₹3.394 = ₹1.806 \text{ per km}
$$

He has to cover an extra cost of ₹9540.

So, distance required to recover this:
$$
\frac{9540}{1.806} \approx 5282.4 \text{ km}
$$

Rounded to nearest whole number: ≈ 5282 km

✔ Answer: He must drive about 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 \times 9 = 4 \times 9 = 36 $

Now compute:
- $ x^2 = 4^2 = 16 $
- $ y^2 = 36^2 = 1296 $

So,
$$
x^2 + y^2 = 16 + 1296 = 1312
$$

✔ Answer: 1312

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

✔ Answer: 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 be $ 10a + b $, where:
- $ a $ = tens digit
- $ b $ = units digit

Given:
1. $ a + b = 9 $ → (Equation 1)
2. $ (10a + b) - 27 = 10b + a $ → (Digits reversed)

Simplify second equation:
$$
10a + b - 27 = 10b + a \\
10a - a + b - 10b = 27 \\
9a - 9b = 27 \\
a - b = 3 \quad \text{(Equation 2)}
$$

Now solve:
From (1): $ a + b = 9 $
From (2): $ a - b = 3 $

Add:
$$
2a = 12 \Rightarrow a = 6 \\
\Rightarrow b = 9 - 6 = 3
$$

So, the number is $ 10a + b = 60 + 3 = 63 $

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

✔ Answer: 63

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(6) If

- $ a + b = 13 $
- $ b + c = 24 $
- $ c + a = 19 $

Find $ a + b + c $?

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

✔ 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 = $ x - 0.12x = 0.88x $
- Spent 8% of remaining: $ 0.08 \times 0.88x = 0.0704x $
- So, left = $ 0.88x - 0.0704x = 0.8096x $

Given that this equals ₹140:
$$
0.8096x = 140 \\
x = \frac{140}{0.8096} \approx 172.925
$$

So, total salary ≈ ₹172.925 → Rs.172.925

Look at options:
a. Rs.1400
b. Rs.17.2925
c. Rs.172.925
d. none of these

✔ 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 $

We know:
1. $ h + d = 20 $ → (1)
2. $ 0.80h + 1.50d = 23.70 $ → (2)

Multiply equation (2) by 10 to eliminate decimals:
$$
8h + 15d = 237 \quad \text{(Equation 2')}
$$

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

Substitute into (2'):
$$
8h + 15(20 - h) = 237 \\
8h + 300 - 15h = 237 \\
-7h = 237 - 300 = -63 \\
h = \frac{-63}{-7} = 9
$$

So, hens = 9, ducks = 11

Check:
- Discount: $ 9 \times 0.80 = 7.20 $
- $ 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) | ~5282 km |
| (3) | 1312 |
| (4) | 13 |
| (5) | 63 |
| (6) | 28 |
| (7) | c. Rs.172.925 |
| (8) | b. 9 |

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