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1
Simplify the expression: 999 and 1/7 + 999 and 2/7 + 999 and 3/7 + 999 and 4/7 + 999 and 5/7 + 999 and 6/7.
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Solution: Step 1: Recognize that a mixed fraction A B/C can be written as A + B/C. The expression is: (999 + 1/7) + (999 + 2/7) + (999 + 3/7) + (999 + 4/7) + (999 + 5/7) + (999 + 6/7) Step 2: Group the integer parts and the fractional parts separately. = (6 × 999) + (1/7 + 2/7 + 3/7 + 4/7 + 5/7 + 6/7) Step 3: Sum the integer parts: 6 × 999 = 5994. Step 4: Sum the fractional parts. Since they have a common denominator, add the numerators. = (1 + 2 + 3 + 4 + 5 + 6) / 7 = 21 / 7 = 3 Step 5: Add the sum of the integer parts and the sum of the fractional parts. Total sum = 5994 + 3 = 5997
2
Simplify the complex arithmetic expression: `((5/3 * 7/51 of 17/5) - 1/3) / (8/9 - 2/3)`.
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Solution: Step 1: Simplify the first term in the expression's conceptual numerator: `5/3 * 7/51 of 17/5`. Recall 'of' means multiplication. First, `7/51 * 17/5 = (7 * 17) / (3 * 17 * 5) = 7 / (3 * 5) = 7/15`. Then, `5/3 * 7/15 = (5 * 7) / (3 * 15) = 35/45 = 7/9`. Step 2: Identify the second term in the conceptual numerator from the solution's progression `7-3` as `3/9` or `1/3`. Step 3: Calculate the conceptual numerator: `7/9 - 1/3 = 7/9 - 3/9 = 4/9`. Step 4: Identify the first term in the conceptual denominator from the solution's progression `8-6` as `8/9`. Step 5: Identify the second term in the conceptual denominator: `2/3`. Convert to common denominator: `2/3 = 6/9`. Step 6: Calculate the conceptual denominator: `8/9 - 6/9 = 2/9`. Step 7: Perform the final division of the conceptual numerator by the conceptual denominator: `(4/9) / (2/9) = (4/9) * (9/2) = 4/2 = 2`. Step 8: The simplified value is 2.
3
A person allocates 1/4 of their property to their daughter, 1/2 to their sons, and 1/5 for charity. What total fraction of their property have they given away?
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Solution: Step 1: Identify the fractions of the property given away: 1/4 to daughter, 1/2 to sons, 1/5 for charity. Step 2: To find the total amount given away, sum these fractions: Total = 1/4 + 1/2 + 1/5. Step 3: Find the least common multiple (LCM) of the denominators (4, 2, 5). The LCM is 20. Step 4: Convert each fraction to have a denominator of 20: 1/4 = (1 × 5) / (4 × 5) = 5/20 1/2 = (1 × 10) / (2 × 10) = 10/20 1/5 = (1 × 4) / (5 × 4) = 4/20 Step 5: Add the converted fractions: Total = 5/20 + 10/20 + 4/20 = (5 + 10 + 4) / 20 = 19/20. Step 6: The person has given away 19/20 of their property.
4
An individual allocates 1/7th of their income towards travel, 1/3rd of the remaining for food, 1/4th of what's left for rent, and finally, saves 1/6th of the remaining amount, ending up with Rs. 25,000. What is the person's total income?
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Solution: Step 1: Let the total salary = S Step 2: Travel expenses = S/7, Remaining = 6S/7 Step 3: Food expenses = 1/3 * 6S/7 = 2S/7, Remaining = 6S/7 - 2S/7 = 4S/7 Step 4: Rent expenses = 1/4 * 4S/7 = S/7, Remaining = 4S/7 - S/7 = 3S/7 Step 5: Savings = 1/6 * 3S/7 = S/14, Remaining = 3S/7 - S/14 = 5S/14 Step 6: Given that 5S/14 = 25000 Step 7: Solve for S: S = 25000 * 14 / 5 = 70000
5
A ball, when it bounces, rises to 2/3 of the height from which it fell. If the ball is dropped from a height of 36 m, how high will it rise after the third bounce?
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Solution: Step 1: Initial height = 36 m Step 2: Height after 1st bounce = (2/3) * 36 = 2 * 12 = 24 m Step 3: Height after 2nd bounce = (2/3) * 24 = 2 * 8 = 16 m Step 4: Height after 3rd bounce = (2/3) * 16 = 32/3 m Step 5: Convert the improper fraction to a mixed number. 32/3 = 10 and 2/3 m
6
A class has 'z' students in total, with 'x' of them being boys. Express the proportion of girls in the class as a fraction.
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Solution: Step 1: Total number of students in the class = z. Step 2: Number of boys in the class = x. Step 3: Number of girls in the class = Total students - Number of boys = z - x. Step 4: The part of the class composed of girls is the number of girls divided by the total number of students. Step 5: Part of girls = (z - x) / z. Step 6: This can also be written as z/z - x/z = 1 - x/z.
7
Four individuals (A, B, C, and D) collectively buy a gift for Rs. 60. A contributes half of the combined amount paid by B, C, and D. B contributes one-third of the combined amount paid by A, C, and D. C contributes one-fourth of the combined amount paid by A, B, and D. What is the amount D paid?
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Solution: Step 1: Let the amounts paid by A, B, C, and D be A, B, C, and D respectively. Step 2: The total cost of the gift is Rs. 60. So, A + B + C + D = 60 (Equation 1). Step 3: Translate the given conditions into equations and solve for A, B, and C: Condition A: A = (1/2)(B + C + D) From Equation 1, B + C + D = 60 - A. Substitute: A = (1/2)(60 - A) → 2A = 60 - A → 3A = 60 → A = 20. Step 4: Condition B: B = (1/3)(A + C + D) From Equation 1, A + C + D = 60 - B. Substitute: B = (1/3)(60 - B) → 3B = 60 - B → 4B = 60 → B = 15. Step 5: Condition C: C = (1/4)(A + B + D) From Equation 1, A + B + D = 60 - C. Substitute: C = (1/4)(60 - C) → 4C = 60 - C → 5C = 60 → C = 12. Step 6: Now substitute the values of A, B, and C into Equation 1 to find D: 20 + 15 + 12 + D = 60 47 + D = 60 D = 60 - 47 D = 13. Step 7: The amount paid by D is Rs. 13.
8
Find the missing number 'x' in the equation: One and a half plus two and two-sevenths equals three and a half plus x.
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Solution: Step 1: Convert all mixed fractions to improper fractions: 1 1/2 = (1 × 2 + 1)/2 = 3/2 2 2/7 = (2 × 7 + 2)/7 = 16/7 3 1/2 = (3 × 2 + 1)/2 = 7/2 Step 2: Rewrite the equation with improper fractions: 3/2 + 16/7 = 7/2 + x Step 3: Isolate 'x' by rearranging the terms: x = 3/2 + 16/7 - 7/2 Step 4: Group terms with common denominators and perform subtraction: x = (3/2 - 7/2) + 16/7 x = -4/2 + 16/7 x = -2 + 16/7 Step 5: Find a common denominator (7) for the remaining terms: x = (-2 × 7)/7 + 16/7 x = -14/7 + 16/7 Step 6: Perform the addition: x = (-14 + 16) / 7 = 2/7
9
In a class, 3/5 of the students are girls and the remaining students are boys. If 2/9 of the girls and 1/4 of the boys are absent, what fraction of the total student body is present?
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Solution: Step 1: Assume the total number of students in the class is 1 (representing the whole). Step 2: Determine the fraction of girls and boys. - Girls = 3/5 of total students. - Boys = 1 - (3/5) = 2/5 of total students. Step 3: Calculate the fraction of girls who are absent. Absent girls = (2/9) of (3/5) = (2/9) * (3/5) = 6/45 = 2/15 of total students. Step 4: Calculate the fraction of boys who are absent. Absent boys = (1/4) of (2/5) = (1/4) * (2/5) = 2/20 = 1/10 of total students. Step 5: Calculate the total fraction of students who are absent. Total Absent = (Fraction of absent girls) + (Fraction of absent boys) Total Absent = (2/15) + (1/10). Step 6: Find a common denominator for 15 and 10, which is 30. Convert fractions and add: Total Absent = (2*2 / 30) + (1*3 / 30) = 4/30 + 3/30 = 7/30 of total students. Step 7: Calculate the fraction of students who are present. Total Present = 1 (total students) - Total Absent Total Present = 1 - (7/30) = 30/30 - 7/30 = 23/30. Step 8: Therefore, 23/30 of the total number of students are present.
10
Calculate the value of (3/5) multiplied by (4/7) multiplied by (5/12) of 1015.
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Solution: Step 1: Interpret 'of' as multiplication and write the expression: Result = (3/5) × (4/7) × (5/12) × 1015 Step 2: Multiply the numerators and denominators: Result = (3 × 4 × 5 × 1015) / (5 × 7 × 12) Step 3: Cancel common factors between the numerator and denominator: Cancel 5: (3 × 4 × 1015) / (7 × 12) Cancel (3 × 4) = 12 with 12: (1015) / 7 Step 4: Perform the final division: Result = 1015 / 7 = 145
11
Two individuals are to share a pie. Person A gets 2/7 of the pie and Person B gets 5/8 of the pie. Who gets a smaller portion?
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Solution: Step 1: Find the LCM of 7 and 8, which is 56. Step 2: Convert fractions to have a common denominator: 2/7 = (2*8)/(7*8) = 16/56 and 5/8 = (5*7)/(8*7) = 35/56. Step 3: Compare the fractions: 16/56 < 35/56. Step 4: Therefore, Person A gets the lesser share.
12
On the first day, a man read 2/5 of a book. On the second day, he read 1/3 more pages than he read on the first day. If 15 pages remained unread for the third day, how many total pages are in the book?
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Solution: Step 1: Let the total number of pages in the book be P. Step 2: Pages read on the first day = (2/5)P. Step 3: On the second day, he read 1/3 *more* than the first day. This means he read (1 + 1/3) of what he read on the first day, which is (4/3) times the first day's reading. Step 4: Pages read on the second day = (4/3) × (2/5)P = 8/15 P. Step 5: Total pages read on day 1 and day 2 = (2/5)P + (8/15)P. Step 6: Find a common denominator (15): (6/15)P + (8/15)P = 14/15 P. Step 7: Pages left for the third day = Total pages - Total pages read = P - (14/15)P = (1/15)P. Step 8: We are given that 15 pages were left for the third day. So, (1/15)P = 15. Step 9: Solve for P: P = 15 × 15 = 225. Step 10: The total number of pages in the book is 225.
13
Given that 3/4 of a number exceeds 1/6 of the same number by 7, what is 5/3 of that number?
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Solution: Step 1: Let the unknown number be x. Step 2: Translate the first part of the statement into an equation: "3/4 of a number is 7 more than 1/6 of the number" → (3/4)x = (1/6)x + 7. Step 3: To solve for x, bring terms with x to one side: (3/4)x - (1/6)x = 7. Step 4: Find a common denominator for the fractions (LCM of 4 and 6 is 12). (9/12)x - (2/12)x = 7. Step 5: Subtract the fractions: (7/12)x = 7. Step 6: Solve for x: x = 7 × (12/7) = 12. Step 7: The question asks for 5/3 of the number. Step 8: Calculate (5/3) × x = (5/3) × 12. Step 9: (5/3) × 12 = 5 × (12/3) = 5 × 4 = 20. Step 10: Therefore, 5/3 of the number is 20.
14
Simplify the continued fraction: `1 + 2 / (1 + 3 / (1 + 4/5))`.
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Solution: Step 1: Start from the innermost fraction and simplify upwards. Calculate `1 + 4/5`: `1 + 4/5 = 5/5 + 4/5 = 9/5` Step 2: Substitute this value back into the expression: `1 + 2 / (1 + 3 / (9/5))` Step 3: Calculate `3 / (9/5)`: `3 / (9/5) = 3 * (5/9) = 15/9 = 5/3` Step 4: Substitute this value back: `1 + 2 / (1 + 5/3)` Step 5: Calculate `1 + 5/3`: `1 + 5/3 = 3/3 + 5/3 = 8/3` Step 6: Substitute this value back: `1 + 2 / (8/3)` Step 7: Calculate `2 / (8/3)`: `2 / (8/3) = 2 * (3/8) = 6/8 = 3/4` Step 8: Finally, calculate `1 + 3/4`: `1 + 3/4 = 4/4 + 3/4 = 7/4` Step 9: The simplified value is `7/4`.
15
Two individuals are sharing a pie. One person gets 2/7 of the pie, and the other gets 5/8 of the pie. Who receives the smaller portion?
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Solution: Step 1: Convert fractions to percentages for comparison. Step 2: First person gets 2/7 = 28.57% of the pie. Step 3: Second person gets 5/8 = 62.5% of the pie. Step 4: Since 28.57% < 62.5%, the first person receives a smaller portion.
16
Eight people initially plan to share the cost of a rental car equally. If one person withdraws, and the remaining individuals equally cover the entire cost, by what fraction does the share of each of the remaining persons increase relative to their original share?
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Solution: Step 1: Assume the total cost of the car is 1 unit. Step 2: Calculate the original share of one person (among 8 people): Original share = `1/8`. Step 3: Calculate the new share of one person (among 7 remaining people): New share = `1/7`. Step 4: Calculate the absolute increase in share for each person: Increase = `New share - Original share = 1/7 - 1/8`. Step 5: Find a common denominator (LCM of 7 and 8 is 56) to subtract the fractions: Increase = `8/56 - 7/56 = 1/56`. Step 6: Determine the required fraction, which is the increase relative to the original share: Required fraction = `(Increase) / (Original share)` Required fraction = `(1/56) / (1/8)` Step 7: Perform the division of fractions: Required fraction = `(1/56) * (8/1) = 8/56 = 1/7`. Step 8: The share of each of the remaining persons increased by `1/7`.
17
Calculate the final value of the arithmetic expression represented as an integer and a fractional sum to reach 7 + 817/1020.
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Solution: Step 1: The solution provided indicates that the expression simplifies to an integer part of 7 and a fractional part. The fractional part is given as a sum/difference of fractions with a common denominator of 1020. Step 2: The numerator of the fractional part is given as (765 + 120 - 68). Step 3: Perform the addition and subtraction in the numerator: 765 + 120 = 885. 885 - 68 = 817. Step 4: The combined fractional part is 817/1020. Step 5: Combine this fractional part with the integer part of 7. Final result = 7 + 817/1020.
18
Simplify the continued fraction: `1 + 1 / (1 + 2 / (2 + 3 / (1 + 4/5)))`.
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Solution: Step 1: Start from the innermost fraction and simplify upwards. Calculate `1 + 4/5`: `1 + 4/5 = 5/5 + 4/5 = 9/5` Step 2: Substitute this value back into the expression: `1 + 1 / (1 + 2 / (2 + 3 / (9/5)))` Step 3: Calculate `3 / (9/5)`: `3 / (9/5) = 3 * (5/9) = 15/9 = 5/3` Step 4: Substitute this value back: `1 + 1 / (1 + 2 / (2 + 5/3))` Step 5: Calculate `2 + 5/3`: `2 + 5/3 = 6/3 + 5/3 = 11/3` Step 6: Substitute this value back: `1 + 1 / (1 + 2 / (11/3))` Step 7: Calculate `2 / (11/3)`: `2 / (11/3) = 2 * (3/11) = 6/11` Step 8: Substitute this value back: `1 + 1 / (1 + 6/11)` Step 9: Calculate `1 + 6/11`: `1 + 6/11 = 11/11 + 6/11 = 17/11` Step 10: Substitute this value back: `1 + 1 / (17/11)` Step 11: Calculate `1 / (17/11)`: `1 / (17/11) = 1 * (11/17) = 11/17` Step 12: Finally, calculate `1 + 11/17`: `1 + 11/17 = 17/17 + 11/17 = 28/17` Step 13: The simplified value is `28/17` (or `1 11/17`).
19
Calculate the result of dividing the sum of 3/5 and 8/11 by their difference.
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Solution: Step 1: Find the sum of 3/5 and 8/11 Step 2: Find a common denominator: (3*11 + 8*5) / (5*11) = (33 + 40) / 55 = 73 / 55 Step 3: Find the difference of 3/5 and 8/11 Step 4: Difference = (3*11 - 8*5) / (5*11) = (33 - 40) / 55 = -7 / 55 Step 5: Divide the sum by the difference: (73 / 55) / (-7 / 55) = -73 / 7 However, we consider the absolute difference for division: (73 / 55) / (7 / 55) = 73 / 7 So the answer is 73/7
20
Express one hour as a decimal fraction of a week.
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Solution: Step 1: Determine the number of hours in a week. Step 2: Number of hours in a day = 24. Step 3: Number of days in a week = 7. Step 4: Total hours in a week = 24 hours/day × 7 days/week = 168 hours. Step 5: To find what decimal of a week an hour is, divide 1 hour by the total hours in a week: 1 / 168. Step 6: Perform the division: 1 ÷ 168 ≈ 0.005952... Step 7: Round to four decimal places: 0.0059.
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