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0.1 of a bamboo's length is in mud, and 5/8 of its length is in water. If the remaining 2.75 meters are above water, what is the total length of the bamboo?
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Solution: Step 1: Convert the decimal part to a fraction. 0.1 = 1/10 Step 2: Calculate the total fraction of the bamboo that is in mud and water. Fraction in mud and water = 1/10 + 5/8 Step 3: Find a common denominator for the fractions (LCM of 10 and 8 is 40). Fraction in mud and water = (4/40) + (25/40) = 29/40 Step 4: Determine the fraction of the bamboo that is above water. Fraction above water = Total length (1) - Fraction in mud and water Fraction above water = 1 - 29/40 = (40 - 29) / 40 = 11/40 Step 5: Use the given length of the part above water to find the total length. Let the total length of the bamboo be L. (11/40) × L = 2.75 m L = 2.75 × (40/11) L = (275/100) × (40/11) L = (11 × 25 / 100) × (40/11) L = (25/100) × 40 L = (1/4) × 40 L = 10 m. The total length of the bamboo is 10 meters.
2
By how much does the value of (6 × 8/7) exceed the value of (6 / (7 × 8))?
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Solution: Step 1: Calculate the value of the first expression: (6 × 8/7). 6 × 8/7 = 48/7. Step 2: Calculate the value of the second expression: (6 / (7 × 8)). 6 / (7 × 8) = 6 / 56. Step 3: Simplify the second fraction. 6/56 = 3/28. Step 4: Find the difference between the first value and the second value. Difference = 48/7 - 3/28. Step 5: Find a common denominator for the fractions (LCM of 7 and 28 is 28). 48/7 = (48 × 4) / (7 × 4) = 192/28. Step 6: Perform the subtraction. Difference = 192/28 - 3/28 = (192 - 3) / 28 = 189/28. Step 7: Simplify the resulting fraction and convert to a mixed number. 189/28 = 27/4. 27/4 = 6 with a remainder of 3, so 6 3/4. The first value exceeds the second by 6 3/4.
3
A runner has completed 1 1/4 laps of a 5-lap race. How many laps are left to run?
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Solution: Step 1: Identify the total length of the race and the distance already run. Total race length = 5 laps Distance run = 1 1/4 laps Step 2: Convert the mixed number to an improper fraction. 1 1/4 = (1 × 4 + 1) / 4 = 5/4 laps Step 3: Subtract the distance run from the total race length to find the remaining distance. Remaining laps = Total laps - Laps run Remaining laps = 5 - 5/4 Step 4: Perform the subtraction. To subtract, find a common denominator: 5 = 20/4 Remaining laps = 20/4 - 5/4 Remaining laps = (20 - 5) / 4 Remaining laps = 15/4 laps. The remaining laps to be run are 15/4, which can also be expressed as 3 3/4 laps.
4
A rod is colored as follows: 1/10 is red, 1/20 is orange, 1/30 is yellow, 1/40 is green, 1/50 is blue, 1/60 is black. The remaining portion is violet, measuring 12.08 meters. What is the total length of the rod?
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Solution: Step 1: Calculate the total fraction of the rod covered by the specified colors. Total colored fraction = 1/10 + 1/20 + 1/30 + 1/40 + 1/50 + 1/60 Step 2: Find the Least Common Multiple (LCM) of the denominators (10, 20, 30, 40, 50, 60). LCM(10, 20, 30, 40, 50, 60) = 600. Step 3: Convert each fraction to have the common denominator and sum them. (60/600) + (30/600) + (20/600) + (15/600) + (12/600) + (10/600) = (60 + 30 + 20 + 15 + 12 + 10) / 600 = 147 / 600 Step 4: Calculate the fraction of the rod that is violet. Violet fraction = Total length (1) - Total colored fraction Violet fraction = 1 - 147/600 = (600 - 147) / 600 = 453/600 Step 5: Use the given length of the violet part to find the total length of the rod. Let the total length of the rod be L. (453/600) × L = 12.08 m L = 12.08 × (600 / 453) L = 12.08 × (200 / 151) (by dividing 600 and 453 by their GCD, 3) L = (1208 / 100) × (200 / 151) L = (1208 / 151) × (200 / 100) L = 8 × 2 (since 1208 / 151 = 8) L = 16 m. The total length of the rod is 16 meters.
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