Finding the Unknown
1. What is an equation? Give one example.
Answer:
An equation is a statement of equality between two algebraic expressions. It shows that the value of the Left Hand Side (LHS) is equal to the value of the Right Hand Side (RHS).
Example:
\(3x + 4 = 7\)
Here:
LHS = \(3x + 4\)
RHS = \(7\)
The equation is true only for the value of x that makes both sides equal.
2. In a matchstick pattern, the number of matchsticks at position n is given by \(2n+1\). Find the position number that contains 99 matchsticks.
Answer:
Given:
\(2n + 1 = 99\)Subtract 1 from both sides:
\(2n = 98\)Divide both sides by 2:
\(n = 49\)Therefore, the arrangement with 99 matchsticks is at position 49.
3. Why does a balanced weighing scale remain balanced when equal weights are removed from both sides?
Answer:
A balanced weighing scale has equal weight on both sides.
If the same weight is removed from both sides, equality remains unchanged because both sides decrease by the same amount.
This idea is similar to solving equations, where the same operation is performed on both sides without changing the equality.
4. What is meant by solving an equation?
Answer:
Solving an equation means finding the value of the unknown letter-number that makes the LHS equal to the RHS.
For example:
Subtract 5 from both sides:
\(x = 7\)Thus, 7 is the solution because:
\(7 + 5 = 12\)5. Solve the equation:
\(5x – 4 = 11\)
Answer:
Add 4 to both sides:
\(5x = 15\)Divide both sides by 5:
\(x = 3\)Check:
\(5(3) – 4 = 15 – 4 = 11\)LHS = RHS
Therefore, x = 3.
6. Explain why addition and subtraction are called inverse operations.
Answer:
Addition and subtraction undo each other.
Example:
\(15 + 8 = 23\)To remove 8, subtract 8:
\(23 – 8 = 15\)Since one operation reverses the effect of the other, they are called inverse operations.
This idea helps in solving equations.
7. Solve the equation:
\(11y + (-5) = 61\)
Answer:
Add 5 to both sides:
\(11y = 66\)Divide both sides by 11:
\(y = 6\)Check:
\(11(6) + (-5) = 66 – 5 = 61\)LHS = RHS
Therefore, y = 6.
8. Solve the equation:
\(6y + 7 = 4y + 21\)
Answer:
Subtract \(4y\) from both sides:
\(2y + 7 = 21\)Subtract 7 from both sides:
\(2y = 14\)Divide both sides by 2:
\(y = 7\)Therefore, y = 7.
9. Solve the equation:
\(\frac{u}{15}=6\)
Answer:
Multiply both sides by 15:
\(u = 6 \times 15\) \(u = 90\)Therefore, u = 90.
10. A tile pattern follows the rule \(3k+1\). How many tiles are needed in Step 20?
Answer:
Substitute \(k=20\):
\(3(20)+1\) \(=60+1\) \(=61\)Therefore, Step 20 requires 61 tiles.
11. Madhubanti spends ₹25 per snack plate and ₹50 as a delivery charge. If she has ₹500, how many plates can she buy?
Answer:
First subtract the delivery charge:
\(500 – 50 = 450\)Money left for snacks = ₹450
Each plate costs ₹25:
\(450 \div 25 = 18\)Therefore, she can buy 18 plates of snacks.
12. Jahnavi has ₹4000 and saves ₹650 per month. Sunita has ₹5050 and saves ₹500 per month. After how many months will their savings be equal?
Answer:
Equation:
\(4000 + 650m = 5050 + 500m\)Subtract \(500m\):
\(4000 + 150m = 5050\)Subtract 4000:
\(150m = 1050\)Divide by 150:
\(m = 7\)Therefore, their savings become equal after 7 months.
13. Solve:
\(28(x+4)+300=1000\)
Answer:
Subtract 300:
\(28(x+4)=700\)Divide by 28:
\(x+4=25\)Subtract 4:
\(x=21\)Therefore, x = 21.
14. Akash follows a number trick. After all operations, the final answer is 24. Find the starting number.
Operations:
Think of a number
Subtract 3
Multiply by 4
Add 8
Answer:
Let the number be x.
After subtracting 3:
\(x-3\)Multiply by 4:
\(4(x-3)=4x-12\)Add 8:
\(4x-4\)Given final answer = 24
\(4x-4=24\)Add 4:
\(4x=28\)Divide by 4:
\(x=7\)Therefore, the starting number was 7.
15. Ramesh and Suresh have 60 marbles altogether. Ramesh has 30 more marbles than Suresh. How many marbles does each have?
Answer:
Let Suresh have y marbles.
Then Ramesh has:
\(y+30\)Total marbles:
\(y+(y+30)=60\) \(2y+30=60\)Subtract 30:
\(2y=30\)Divide by 2:
\(y=15\)Suresh has 15 marbles.
Ramesh has:
\(15+30=45\)Answer:
Suresh = 15 marbles
Ramesh = 45 marbles
16. State the three important observations used while solving equations.
Answer:
When a term is removed from one side, its additive inverse appears on the other side.
When a factor is removed from one side, the other side is divided by that factor.
When a divisor is removed from one side, the other side is multiplied by that divisor.
These rules help solve equations systematically.
17. Give a real-life situation represented by the equation:
\(100x+75=250\)
Answer:
Suppose:
A snack plate costs ₹x.
100 plates are bought.
Delivery charge is ₹75.
Total cost is ₹250.
Then:
\(100x+75=250\)This equation models the total expense.
18. A brick weighs 1 kg more than half its weight. Find the weight of the brick.
Answer:
Let the weight be x kg.
According to the statement:
\(x=\frac{x}{2}+1\)Subtract \(\frac{x}{2}\):
\(\frac{x}{2}=1\)Multiply by 2:
\(x=2\)Therefore, the brick weighs 2 kg.
19. One quarter of a number increased by 9 equals the number itself. Find the number.
Answer:
Let the number be x.
\(\frac{x}{4}+9=x\)Subtract \(\frac{x}{4}\):
\(9=\frac{3x}{4}\)Multiply by 4:
\(36=3x\)Divide by 3:
\(x=12\)Therefore, the number is 12.
20. Why is algebra called a powerful language of mathematics?
Answer:
Algebra uses letters and symbols to represent unknown quantities. It helps us:
Find unknown values.
Describe patterns.
Solve real-life problems.
Write general rules.
Justify mathematical statements.
Because it can represent many situations in a simple and precise way, algebra is called a powerful language of mathematics.
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