If \(1^2 + 2^2 + 3^2 + ... + 10^2 = 385\), then the value of \(2^2 + 4^2 + 6^2 + ... + 20^2\) is:
Answer: C
Let the required sum be S.
\(S = 2^2 + 4^2 + 6^2 + ... + 20^2\).
\(S = (2 \times 1)^2 + (2 \times 2)^2 + (2 \times 3)^2 + ... + (2 \times 10)^2\).
\(S = 2^2(1^2) + 2^2(2^2) + 2^2(3^2) + ... + 2^2(10^2)\).
\(S = 2^2 (1^2 + 2^2 + 3^2 + ... + 10^2)\).
\(S = 4 \times (\text{given sum})\).
\(S = 4 \times 385 = 1540\).
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A number whose fifth part increased by 4 is equal to its fourth part diminished by 10. The number is:
Answer: C
Let the number be x.
The equation is \(\frac{x}{5} + 4 = \frac{x}{4} - 10\).
\(4 + 10 = \frac{x}{4} - \frac{x}{5}\).
\(14 = \frac{5x - 4x}{20}\).
\(14 = \frac{x}{20}\).
\(x = 14 \times 20 = 280\).
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The sum of all two-digit numbers divisible by 5 is:
Answer: C
The two-digit numbers divisible by 5 are 10, 15, 20, ..., 95. This is an arithmetic progression.
First term (a) = 10. Last term (l) = 95. Common difference (d) = 5.
Number of terms (n) = \((\frac{l-a}{d}) + 1 = (\frac{95-10}{5}) + 1 = 17 + 1 = 18\).
Sum = \(\frac{n}{2}(a+l) = \frac{18}{2}(10+95) = 9 \times 105 = 945\).
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If the sum of two numbers is 22 and the sum of their squares is 404, then the product of the numbers is:
Answer: A
Let the numbers be a and b. We have a+b=22 and a²+b²=404.
We use the identity \((a+b)² = a² + b² + 2ab\).
\(22² = 404 + 2ab\)
484 = 404 + 2ab
80 = 2ab
ab = 40.
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A number is doubled and 9 is added. If the resultant is trebled, it becomes 75. What is the number?
Answer: B
Let the number be x. We work backward from the result.
Before being trebled, the number was 75 / 3 = 25.
This resultant (25) was obtained after doubling the number and adding 9. So, 2x + 9 = 25.
2x = 25 - 9 = 16.
x = 16 / 2 = 8.
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If a number is decreased by 4 and divided by 6, the result is 8. What would be the result if 2 is subtracted from the number and then it is divided by 5?
Answer: B
Let the number be x.
According to the first condition: \(\frac{x-4}{6} = 8\).
\(x-4 = 48\), so \(x = 52\).
Now, according to the second condition, subtract 2 from the number: \(52 - 2 = 50\).
Then divide by 5: \(50 \div 5 = 10\).
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Find the value of \(0.5 \times 0.05 \times 500\).
Answer: B
We can multiply sequentially.
\(0.5 \times 0.05 = 0.025\).
Now, \(0.025 \times 500 = 25 \times 0.5 = 12.5\).
Alternatively, write as fractions: \((\frac{1}{2}) \times (\frac{5}{100}) \times 500 = (\frac{1}{2}) \times (\frac{1}{20}) \times 500 = \frac{500}{40} = \frac{50}{4} = 12.5\).
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What is the smallest number that must be added to 2203 to make it a perfect square?
Answer: C
We need to find the nearest perfect square greater than 2203.
We know \(40^2 = 1600\) and \(50^2 = 2500\). The root is between 40 and 50.
Let's try 47: \(47^2 = 2209\).
The next perfect square after 2203 is 2209.
The number to be added is \(2209 - 2203 = 6\).
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The sum of the digits of a 2-digit number is 9. When 27 is added to the number, the digits get reversed. The number is:
Answer: B
Let the ten's digit be x and the unit's digit be y. Number = 10x + y.
Given: x + y = 9.
Given: (10x + y) + 27 = 10y + x (reversed number).
\(9x - 9y = -27\), which simplifies to \(x - y = -3\) or \(y - x = 3\).
We have two equations: x + y = 9 and y - x = 3.
Adding them: 2y = 12, so y = 6.
Substituting y=6 into x+y=9, we get x=3.
The number is 36.
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What is 25% of 50% of 200?
Answer: B
First, calculate 50% of 200.
50% of 200 = \(0.5 \times 200 = 100\).
Now, calculate 25% of 100.
25% of 100 = 25.
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