Question

mixed math problem set
10 problems - easy, medium & hard

easy

1. evaluate: (47 + 68 - 25).
2. a shirt costs \\$24. what is the total price for 3 shirts?
3. what is (15\\%) of 200?

medium

1. solve for (x): (3x - 7 = 2x + 5).
2. a rectangle has a length of 12 cm and a width of 7 cm. find its area and perimeter.
3. the average of five numbers is 18. four of them are 12, 20, 15, and 25. find the fifth number.
4. simplify: ((2^3 \times 2^4) / 2^2).

hard

1. solve the quadratic equation: (x^2 - 5x + 6 = 0).
2. a train travels 240 km in 3 hours, then 180 km in 2 hours. what is its average speed for the whole journey?
3. find the value of (x): (\log_2(x) + \log_2(x - 2) = 3).

Explanation:

🆕 New Concept Discovered: Multi-Topic Math Review
Solving a diverse set of fundamental math problems.

Step 1: Solve Easy Problems (1–3)

• Problem 1: Evaluate \( 47 + 68 - 25 \).

First, add the first two numbers:

$$ 47 + 68 = 115 $$

Next, subtract 25:

$$ 115 - 25 = 90 $$

• Problem 2: A shirt costs \$24. What is the total price for 3 shirts?

Multiply the cost of one shirt by the quantity:

$$ 24 \times 3 = 72 $$

The total price is \$72.

• Problem 3: What is \( 15\% \) of 200?

Convert the percentage to a decimal and multiply:

$$ 0.15 \times 200 = 30 $$

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Step 2: Solve Medium Problems (4–7)

• Problem 4: Solve for \( x \): \( 3x - 7 = 2x + 5 \).

Subtract \( 2x \) from both sides:

$$ x - 7 = 5 $$

Add 7 to both sides:

$$ x = 12 $$

• Problem 5: A rectangle has a length of \( 12\text{ cm} \) and a width of \( 7\text{ cm} \). Find its area and perimeter.
• Area is length multiplied by width:

$$ \text{Area} = 12 \times 7 = 84\text{ cm}^2 $$

• Perimeter is the total distance around the outside:

$$ \text{Perimeter} = 2 \times (12 + 7) = 2 \times 19 = 38\text{ cm} $$

• Problem 6: The average of five numbers is 18. Four of them are 12, 20, 15, and 25. Find the fifth number.

Let the fifth number be \( y \). The sum of all five numbers is:

$$ 5 \times 18 = 90 $$

Sum the four known numbers:

$$ 12 + 20 + 15 + 25 = 72 $$

Subtract this sum from the total sum to find \( y \):

$$ y = 90 - 72 = 18 $$

• Problem 7: Simplify: \( \frac{2^3 \times 2^4}{2^2} \).

Use exponent rules. When multiplying with the same base, add the exponents:

$$ 2^3 \times 2^4 = 2^{3+4} = 2^7 $$

When dividing with the same base, subtract the exponents:

$$ \frac{2^7}{2^2} = 2^{7-2} = 2^5 = 32 $$

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Step 3: Solve Hard Problems (8–10)

• Problem 8: Solve the quadratic equation: \( x^2 - 5x + 6 = 0 \).

Factor the quadratic expression by finding two numbers that multiply to 6 and add to \(-5\). These numbers are \(-2\) and \(-3\):

$$ (x - 2)(x - 3) = 0 $$

Set each factor to zero:

$$ x = 2 \quad \text{or} \quad x = 3 $$

• Problem 9: A train travels \( 240\text{ km} \) in 3 hours, then \( 180\text{ km} \) in 2 hours. What is its average speed for the whole journey?

Average speed is the total distance divided by the total time:

$$ \text{Total Distance} = 240 + 180 = 420\text{ km} $$

$$ \text{Total Time} = 3 + 2 = 5\text{ hours} $$

$$ \text{Average Speed} = \frac{420}{5} = 84\text{ km/h} $$

• Problem 10: Find the value of \( x \): \( \log_2(x) + \log_2(x - 2) = 3 \).

Use the logarithm product rule, \( \log_b(a) + \log_b(c) = \log_b(a \cdot c) \):

$$ \log_2(x(x - 2)) = 3 $$

Convert the logarithmic equation into its exponential form:

$$ x(x - 2) = 2^3 $$

$$ x^2 - 2x = 8 $$

$$ x^2 - 2x - 8 = 0 $$

Factor the quadratic equation:

$$ (x - 4)(x + 2) = 0 $$

This gives \( x = 4 \) or \( x = -2 \).
Since the argument of a logarithm must be strictly positive (\( x > 0 \) and \( x - 2 > 0 \)), we discard \( x = -2 \). Thus, \( x = 4 \).

Answer:

1. 90
2. \$72
3. 30
4. \( x = 12 \)
5. Area = \( 84\text{ cm}^2 \), Perimeter = \( 38\text{ cm} \)
6. 18
7. 32 (or \( 2^5 \))
8. \( x = 2 \) or \( x = 3 \)
9. \( 84\text{ km/h} \)
10. \( x = 4 \)