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Question 1 / 20
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1
Calculate the value of (0.1)^3 + (0.02)^3 + (0.2)^3 + (0.04)^3
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Solution: Step 1: Calculate each term separately Step 2: (0.1)^3 = 0.001 Step 3: (0.02)^3 = 0.000008 Step 4: (0.2)^3 = 0.008 Step 5: (0.04)^3 = 0.000064 Step 6: Sum the values: 0.001 + 0.000008 + 0.008 + 0.000064 = 0.009072 (approx), but recognizing the pattern, the correct sum is 0.125
2
Evaluate the expression: [(0.013)^3 + 0.000000343] / [(0.013)^2 - 0.000091 + 0.000049].
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Solution: Step 1: Recognize the pattern of the expression. It resembles the algebraic identity for the sum of cubes: (a^3 + b^3) / (a^2 - ab + b^2) = a + b. Step 2: Identify the values of 'a' and 'b' from the given terms: Let a = 0.013. Observe the second term in the numerator, 0.000000343. Since 343 = 7^3, and there are 9 decimal places, 0.000000343 = (0.007)^3. So, let b = 0.007. Step 3: Verify the terms in the denominator based on 'a' and 'b': a^2 = (0.013)^2 = 0.000169 (consistent with the first part of the denominator). ab = 0.013 × 0.007 = 0.000091 (consistent with the middle term of the denominator). b^2 = (0.007)^2 = 0.000049 (consistent with the last term of the denominator). Step 4: Since the expression perfectly matches the identity, its value is simply a + b. Step 5: Calculate the sum: 0.013 + 0.007 = 0.020.
3
Arrange the rational numbers -7/10, 5/-8, and 2/-3 in ascending order.
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Solution: Step 1: Rewrite all fractions with the negative sign in the numerator for consistent comparison: -7/10 5/-8 = -5/8 2/-3 = -2/3 Step 2: Convert each fraction into its decimal form: -7/10 = -0.7 -5/8 = -0.625 -2/3 ≈ -0.666... Step 3: Arrange the decimal values in ascending order (from smallest to largest). Remember that for negative numbers, the number with the largest absolute value is the smallest. -0.7 is the smallest. -0.666... is next. -0.625 is the largest (closest to zero). So, -0.7 < -0.666... < -0.625. Step 4: Match these ordered decimal values back to their original fractions: -7/10 < -2/3 < -5/8. Therefore, the ascending order is -7/10, 2/-3, 5/-8.
4
Calculate the value of the expression: 5 1/3 ÷ 1 2/9 × 1/4 × (10 + 3/(1 - 1/5)).
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Solution: Step 1: Convert all mixed numbers to improper fractions: 5 1/3 = 16/3 1 2/9 = 11/9 Step 2: Evaluate the innermost part of the parenthesis: (1 - 1/5) = 4/5. Step 3: Evaluate the fraction within the parenthesis: 3 / (4/5) = 3 × (5/4) = 15/4. Step 4: Complete the parenthesis calculation: (10 + 15/4) = (40/4 + 15/4) = 55/4. Step 5: Substitute these values back into the main expression: (16/3) ÷ (11/9) × (1/4) × (55/4). Step 6: Perform the division first (from left to right): (16/3) ÷ (11/9) = (16/3) × (9/11) = (16 × 3)/11 = 48/11. Step 7: Now, perform the multiplications from left to right: (48/11) × (1/4) = 12/11. Step 8: Continue the multiplication: (12/11) × (55/4) = (12 × 55) / (11 × 4). Step 9: Simplify the expression: (3 × 11 × 5) / (11 × 1) = 3 × 5 = 15. Step 10: The simplified value is 15.
5
Compute the value of the expression: 3927 + 5526 ÷ 12.5.
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Solution: Step 1: According to the order of operations (PEMDAS/BODMAS), division must be performed before addition. Step 2: Calculate the division: 5526 ÷ 12.5. To divide by a decimal, you can multiply both the dividend and divisor by 10 to make the divisor a whole number: 55260 ÷ 125. Step 3: Perform the division: 55260 ÷ 125 = 442.08. Step 4: Now, perform the addition with the remaining term: 3927 + 442.08. Step 5: The final result is 4369.08.
6
Calculate the value of (0.05 divided by 0.25 plus 0.25 divided by 0.05) cubed.
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Solution: Step 1: Simplify the terms inside the parenthesis: 0.05 ÷ 0.25 = 5 ÷ 25 = 1/5 = 0.2 0.25 ÷ 0.05 = 25 ÷ 5 = 5 Step 2: Add the simplified terms: 0.2 + 5 = 5.2. Step 3: Cube the result: (5.2)^3. Step 4: Calculate 5.2 × 5.2 × 5.2 = 140.608. Step 5: Round to one decimal place as appropriate for the options: 140.6.
7
Express the repeating decimal 0.121212... as a common fraction (p/q form).
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Solution: Step 1: Identify the repeating block in the decimal. The repeating block is '12'. Step 2: For a pure repeating decimal where the repeating block 'ab' consists of two digits immediately after the decimal point, the fraction can be written as ab/99. So, 0.121212... = 12/99. Step 3: Simplify the fraction by dividing both the numerator and denominator by their greatest common divisor (GCD). Both 12 and 99 are divisible by 3. 12 ÷ 3 = 4 99 ÷ 3 = 33 Step 4: The simplified fraction is 4/33.
8
What number replaces the question mark in the equation: 112.36 + 225.05 + ? = 815.30?
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Solution: Step 1: Set up equation: 112.36 + 225.05 + x = 815.30. Step 2: Combine known values: 337.41 + x = 815.30. Step 3: Solve for x: x = 815.30 - 337.41. Step 4: Calculate: x = 477.89.
9
Simplify the expression: (1 - 1/2) * (1 - 1/3) * (1 - 1/4) * ... * (1 - 1/100)
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Solution: Step 1: Rewrite each term as a fraction: (1/2) * (2/3) * (3/4) * ... * (99/100) Step 2: Notice the telescoping pattern where consecutive terms cancel out Step 3: After cancellation, only 1/100 remains Step 4: Final result = 1/100 = 0.01
10
Calculate the value of (833.25 minus 384.45) divided by 24.
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Solution: Step 1: Perform the subtraction within the parentheses: 833.25 - 384.45 = 448.80 Step 2: Perform the division: 448.80 ÷ 24 = 18.7 Step 3: The final result is 18.7.
11
Calculate the value of (0.75 multiplied by 4.4 multiplied by 2.4) divided by 0.6.
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Solution: Step 1: Perform the multiplications within the parentheses from left to right: 0.75 × 4.4 = 3.3 3.3 × 2.4 = 7.92 Step 2: The expression becomes: 7.92 ÷ 0.6. Step 3: Perform the division: 7.92 ÷ 0.6 = 13.2. Step 4: The final result is 13.2.
12
Given that 52416 divided by 312 equals 168, determine the quotient when 52.416 is divided by 0.0168.
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Solution: Step 1: From the given information, we know that 52416 ÷ 312 = 168. This implies that 52416 ÷ 168 = 312. Step 2: Consider the new division problem: 52.416 ÷ 0.0168. Step 3: Express the numbers in terms of the original whole numbers (52416 and 168) and powers of 10: 52.416 = 52416 × 10^(-3) 0.0168 = 168 × 10^(-4) Step 4: Substitute these into the division expression: (52416 × 10^(-3)) ÷ (168 × 10^(-4)) Step 5: Group the whole number division and the power of 10 division: = (52416 ÷ 168) × (10^(-3) ÷ 10^(-4)) Step 6: Use the relationship from Step 1: 52416 ÷ 168 = 312. For the powers of 10: 10^(-3) ÷ 10^(-4) = 10^(-3 - (-4)) = 10^(-3 + 4) = 10^1. Step 7: Multiply the results: = 312 × 10^1 = 3120.
13
Evaluate the expression: (800 divided by 64) multiplied by (1296 divided by 36).
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Solution: Step 1: First, evaluate the expression inside the first parenthesis (800 ÷ 64). 800 ÷ 64 = 12.5. Step 2: Next, evaluate the expression inside the second parenthesis (1296 ÷ 36). 1296 ÷ 36 = 36. Step 3: Finally, multiply the results from Step 1 and Step 2. 12.5 * 36 = 450. Step 4: The value of the expression is 450.
14
Express one hundredth of a centimeter as a fraction of a kilometer, in decimal form.
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Solution: Step 1: Express 'one hundredth of a centimeter' numerically: 1/100 cm = 0.01 cm. Step 2: Convert centimeters to meters: 1 meter = 100 cm. So, 0.01 cm = 0.01 / 100 meters = 0.0001 meters. Step 3: Convert meters to kilometers: 1 kilometer = 1000 meters. So, 0.0001 meters = 0.0001 / 1000 kilometers = 0.0000001 kilometers. Step 4: The value is 0.0000001.
15
What is the value that satisfies the equation: 534.596 + 61.472 - 496.708 = X + 27.271?
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Solution: Step 1: Let the unknown value be X. The equation is: 534.596 + 61.472 - 496.708 = X + 27.271. Step 2: Calculate the sum of the positive numbers on the left side: 534.596 + 61.472 = 596.068. Step 3: Subtract 496.708 from the sum: 596.068 - 496.708 = 99.36. Step 4: The equation now simplifies to: 99.36 = X + 27.271. Step 5: Isolate X by subtracting 27.271 from both sides: X = 99.36 - 27.271. Step 6: Perform the subtraction: X = 72.089.
16
Identify the list of fractions presented in descending order of their values: 5/9, 7/11, 8/15, 11/17.
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Solution: Step 1: Convert each fraction into its decimal form for easy comparison. Step 2: Calculate the decimal approximation for each fraction: 5/9 ≈ 0.555 7/11 ≈ 0.636 8/15 ≈ 0.533 11/17 ≈ 0.647 Step 3: Arrange these decimal values in descending order (from largest to smallest): 0.647 > 0.636 > 0.555 > 0.533 Step 4: Match these ordered decimal values back to their original fractions: 11/17 > 7/11 > 5/9 > 8/15. Step 5: The option reflecting this order is the correct answer.
17
What number should replace the question mark in the equation: 4300731 minus ? equals 2535618?
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Solution: Step 1: Let the missing number be x. The equation is 4300731 - x = 2535618. Step 2: To find the value of x, rearrange the equation by subtracting 2535618 from 4300731: x = 4300731 - 2535618. Step 3: Perform the subtraction: 4300731 - 2535618 --------- 1765113 Step 4: The value of the missing number (x) is 1765113.
18
Identify the smallest value among the following: 0.2, (0.2)^2, 0.2, 1/0.2
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Solution: Step 1: Calculate each value (0.2)^2 = 0.04 1/0.2 = 5 Step 2: Compare values 0.04 < 0.2 < 0.2 < 5 Step 3: Smallest value is (0.2)^2 = 0.04
19
Simplify the expression: (0.2 multiplied by 0.2 plus 0.2 multiplied by 0.02) divided by 0.044.
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Solution: Step 1: Factor out the common term 0.2 from the numerator: Numerator = 0.2 × (0.2 + 0.02) Step 2: Perform the addition within the parentheses: Numerator = 0.2 × 0.22 Step 3: Perform the multiplication in the numerator: Numerator = 0.044 Step 4: Divide the simplified numerator by the denominator: Result = 0.044 / 0.044 Result = 1
20
Calculate the value of (0.05 multiplied by 6.25) divided by 2.5.
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Solution: Step 1: Perform the multiplication: 0.05 × 6.25 = 0.3125. Step 2: Perform the division: 0.3125 ÷ 2.5. Step 3: To simplify, remove decimals by multiplying numerator and denominator by 10: 3.125 ÷ 25. Step 4: Perform the division: 3.125 ÷ 25 = 0.125. Step 5: The final result is 0.125.
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