Question 1

A 70 cm long wire is to be cut into two pieces such that one piece will be $\frac{2}{5}$ as long as the other. How many centimetres will the shorter piece be?

Accepted Answer

Concept & Strategy: When one part is described as a fraction of another part, you can use the Ratio Method. If Piece A is (2/5) of Piece B, the ratio of Piece A to Piece B is 2:5. The total length is divided into proportional parts. Step-by-Step Solution: • Establish the ratio: Let the longer piece be x. The shorter piece is (2/5)x. The ratio of Shorter to Longer = 2 : 5. • Find the total parts: Total parts = 2 + 5 = 7 units. • Determine the value of one unit: We know the total wire length is 70 cm. 7 units = 70 cm 1 unit = 10 cm • Calculate the shorter piece: Shorter piece = 2 units 2 × 10 cm = 20 cm Exam Strategy & Shortcut: You can instantly solve this mentally by realizing the parts sum to 2+5=7. The shorter piece is exactly (2/7) of the *entire* wire. (2/7) × 70 = 20 cm This bypasses setting up variables entirely. Common Pitfall: A common trap is assuming the shorter piece is (2/5) of the *total* wire length (which would yield 70 × (2/5) = 28). Carefully read whether it says "(2/5) of the total" or "(2/5) as long as the *other*". Final Answer: Therefore, the correct answer is 20.

Question 2

$\frac{1}{10}$ of a pole is coloured red, $\frac{1}{20}$ white, $\frac{1}{30}$ blue, $\frac{1}{40}$ black, $\frac{1}{50}$ violet, $\frac{1}{60}$ yellow and the rest is green. If the length of the green portion of the pole is 12.08 metres, then the l…

Accepted Answer

Concept & Strategy: The whole pole represents 1 entire unit. The sum of all the coloured fractions subtracted from 1 will give the fraction representing the green portion. Equating this final fraction to the actual length given (12.08 m) allows you to solve for the total length of the pole. Step-by-Step Solution: • Add all the given fractions: Sum = (1/10) + (1/20) + (1/30) + (1/40) + (1/50) + (1/60) • Find the Least Common Multiple (LCM) of the denominators: The LCM of 10, 20, 30, 40, 50, 60 is 600. • Convert each fraction: Sum = (60 + 30 + 20 + 15 + 12 + 10/600) = (147/600) • Find the remaining (green) fraction: Green Fraction = 1 - (147/600) = (453/600) • Calculate the total length: Let the total length be x. (453/600) × x = 12.08 x = 12.08 × (600/453) x = (1208 × 6/453) x = (7248/453) = 16 m Exam Strategy & Shortcut: When facing a messy division like 7248 \div 453, use unit digits to quickly estimate. The last digit of 453 is 3. We need a multiplier that gives 8 in the unit digit (3 × 6 = 18). Looking at the options (16, 18, 20, 30), only 16 ends in a 6. A quick check of 450 × 16 confirms it is around 7200. Common Pitfall: Making arithmetic mistakes while calculating the LCM for six different numbers. If you miss a factor, the common denominator will be wrong, drastically changing the remainder fraction and the final calculation. Final Answer: Therefore, the correct answer is 16 m.

Question 3

A waiter's salary consists of his salary and tips. During one week his tips were $\frac{5}{4}$ of his salary. What fraction of his income came from tips?

Accepted Answer

Concept & Formula: Total Income consists of basic salary added to the tips received. Total Income = Salary + Tips Required Fraction = \frac{Tips}{Total Income} Step-by-Step Solution: • Set up relative values: Let the basic salary be S. Tips = (5/4)S • Calculate the total income: Total Income = S + (5/4)S = (9/4)S • Find the proportion originating from tips: Fraction from Tips = (\frac{5/4)S}{(9/4)S} = (5/4) × (4/9) = (5/9) Exam Strategy & Shortcut: Substitute a real value to eliminate variables instantly. Let the salary be ₹ 4 (chosen specifically to clear the denominator of (5/4)). • Salary = ₹ 4 • Tips = (5/4) × 4 = ₹ 5 • Total Income = 4 + 5 = ₹ 9 • Fraction from tips = (5/9) Common Pitfall: Inverting the ratio values or calculating the fraction relative to basic salary rather than the *total income*, which leads directly to selecting the distractor option (5/4). Final Answer: Therefore, the correct answer is (5/9).

Question 4

A sum of ₹ 1360 has been divided among $A$, $B$ and $C$ such that $A$ gets $\frac{2}{3}$ of what $B$ gets and $B$ gets $\frac{1}{4}$ of what $C$ gets. B's share is

Accepted Answer

Concept & Formula: Express the shares of A, B, and C as a combined ratio. Given: A = (2/3)B \implies (A/B) = (2/3) B = (1/4)C \implies (B/C) = (1/4) Combine these fractions to get the continuous ratio A : B : C. Step-by-Step Solution: • Align and combine ratios: We have A : B = 2 : 3 and B : C = 1 : 4. To equalize the B term across both ratios, multiply the B : C ratio by 3: B : C = 3 : 12 Thus, the unified ratio is A : B : C = 2 : 3 : 12. • Compute total units: Total Units = 2 + 3 + 12 = 17 units • Equate to total value to find 1 unit: 17 units = 1360 1 unit = (1360/17) = 80 • Find B's share: B = 3 units = 3 × 80 = 240 Exam Strategy & Shortcut: Once you determine the unified ratio components 2:3:12, notice that B's part is 3. The answer must be a multiple of 3. Looking at the options: • 120 (multiple of 3) • 160 (no) • 240 (multiple of 3) • 300 (multiple of 3) Execute the fast division: (1360/17) = 80. Then 3 × 80 = 240. Common Pitfall: Mistaking the final ratio updates or incorrectly equating the fraction parts, yielding an incorrect unit value. Double-check your unified ratio by verifying if B remains (1/4) of C. Here, (3/12) = (1/4), which confirms correctness. Final Answer: Therefore, the correct answer is ₹ 240.

Question 5

Three friends had dinner at a restaurant. When the bill was received, Amita paid $\frac{2}{3}$ as much as Veena paid and Veena paid $\frac{1}{2}$ as much as Tanya paid. What fraction of the bill did Veena pay?

Accepted Answer

Concept & Strategy: Express each person's relative contribution as a single ratio chain, similar to multi-variable component sharing. Let A = Amita, V = Veena, T = Tanya. A = (2/3)V \implies (A/V) = (2/3) V = (1/2)T \implies (V/T) = (1/2) Step-by-Step Solution: • Unify the component ratios: We need V to be identical in both ratios. The common value for 3 and 1 is 3. Multiply the second ratio (V:T = 1:2) by 3: V : T = 3 : 6 Now join them: A : V : T = 2 : 3 : 6 • Calculate the total bill parts: Total Parts = 2 + 3 + 6 = 11 parts • Determine Veena's fraction of the total: Veena's Fraction = \frac{Veena's Parts}{Total Parts} = (3/11) Exam Strategy & Shortcut: Assume a smart concrete value for the final independent variable, Tanya. Let Tanya pay ₹ 6 (the LCM of the denominators 3 and 2). • Tanya = ₹ 6 • Veena = (1/2) × 6 = ₹ 3 • Amita = (2/3) × 3 = ₹ 2 • Total bill = 6 + 3 + 2 = ₹ 11 • Veena's portion = (3/11) This entirely avoids algebraic variables. Common Pitfall: Summing the initial denominators directly without adjusting them to form a cohesive, unified ratio expression. Final Answer: Therefore, the correct answer is (3/11).

Question 6

$\frac{1}{4}$ of a tank holds 135 litres of water. What part of the tank is full if it contains 180 litres of water?

Accepted Answer

Concept & Formula: This follows the rule of Direct Proportion. The volume of water contained varies directly with the fractional part of the tank that is full. \frac{Fraction_1}{Volume_1} = \frac{Fraction_2}{Volume_2} Step-by-Step Solution: • Find the full capacity of the tank: (1/4) of Capacity = 135 litres Total Capacity = 135 × 4 = 540 litres • Calculate the required fraction for 180 litres: Required Fraction = \frac{Current Volume}{Total Capacity} = (180/540) • Simplify the fraction: Required Fraction = (18/54) = (1/3) Exam Strategy & Shortcut: Set up a direct equation to find the solution without computing the total capacity explicitly: Required Fraction = (180/135) × (1/4) Divide 180 and 135 by 45 (45 × 4 = 180, 45 × 3 = 135): Required Fraction = (4/3) × (1/4) = (1/3) Common Pitfall: Inverting the relationship and computing (135/180) × (1/4), which yields (3/16) and creates confusion when it doesn't appear in the options list. Final Answer: Therefore, the correct answer is (1/3).

Question 7

A tank is $\frac{2}{5}$ full. If 16 litres of water is added to the tank, it becomes $\frac{6}{7}$ full. The capacity of the tank is

Accepted Answer

Concept & Formula: This problem tests the application of linear equations to solve fraction-based word problems. The core insight is that the mathematical difference between the two fractions of the tank's capacity corresponds exactly to the physical volume of water added. Let the total capacity of the tank be represented by x. Step-by-Step Solution: • Given: Initial volume = (2/5)x Final volume = (6/7)x Volume added = 16 litres. • Calculation: Set up the equation based on the change in volume: (6/7)x - (2/5)x = 16 • Find a common denominator for 7 and 5, which is 35, to subtract the fractions: (30x/35) - (14x/35) = 16 (16x/35) = 16 • Solve for x by multiplying both sides by (35/16): x = 16 × (35/16) x = 35 Exam Strategy & Shortcut: Use the LCM & Divisibility Method. The total capacity must be a value that interacts cleanly with the denominators given in the problem (5 and 7) to yield whole numbers. The Least Common Multiple (LCM) of 5 and 7 is 35. Looking at the options (28, 32, 35, 42), only 35 is perfectly divisible by 5. You can immediately identify 35 as the highly probable correct answer without doing the full algebra, saving precious seconds. Common Pitfall: A frequent mistake is setting up the equation backwards (subtracting the larger fraction from the smaller one) resulting in a negative capacity. Always double-check which fraction represents the "more full" state to ensure your resulting difference is positive. Final Answer: Therefore, the correct answer is 35 litres.

Question 8

A drum of kerosene is $\frac{3}{4}$ full. When 30 litres of kerosene is drawn from it, it remains $\frac{7}{12}$ full. The capacity of the drum is

Accepted Answer

Concept & Formula: This is a classic algebraic volume problem based on fractional differences. When a specific quantity is removed, the fractional fullness of the container decreases. By equating the difference between the initial and final fractions to the absolute volume removed, we can find the total capacity. Step-by-Step Solution: • Given: Initial fractional volume = (3/4) Final fractional volume = (7/12) Volume removed = 30 litres. Let total capacity be x. • Calculation: Write the equation representing the removal of kerosene: (3/4)x - 30 = (7/12)x • Rearrange the equation to group the x terms on one side: (3/4)x - (7/12)x = 30 • Find a common denominator (which is 12) to evaluate the left side: (9x/12) - (7x/12) = 30 (2x/12) = 30 (1/6)x = 30 • Multiply by 6 to isolate x: x = 180 Exam Strategy & Shortcut: Use the Fractional Equivalence Method. Calculate the difference in fractions first mentally: (3/4) - (7/12) = (9/12) - (7/12) = (2/12) = (1/6). You now know that (1/6) of the drum equals 30 litres. To find the whole drum (1 full unit), simply multiply 30 × 6 = 180. This skips writing out full algebraic steps and saves time. Common Pitfall: Students often struggle with finding the common denominator quickly, leading to calculation errors. Another mistake is misreading "drawn from it" as "added to it," which would lead to incorrect addition instead of subtraction. Final Answer: Therefore, the correct answer is 180 litres.

Question 9

A tin of oil was $\frac{5}{8}$ full. When 10 bottles of oil was taken out and 8 bottles of oil was poured into it, it was $\frac{3}{5}$ full. How many bottles of oil can the tin contain?

Accepted Answer

Concept & Logic: This problem tests the ability to calculate net change and map it to fractional differences in capacity. Instead of working with the extracted and added volumes separately, find the net change in the number of bottles. Equate this net change to the difference between the initial and final fractions of the tin. Let the total capacity in bottles be x. Step-by-Step Solution: • Given: Initial state = (5/8) full. Final state = (3/5) full. Action: Take out 10 bottles, add 8 bottles. • Calculation: Determine the net change in oil volume: Net oil removed = 10 (out) - 8 (in) = 2 bottles. • Set up the equation reflecting this decrease: (5/8)x - 2 = (3/5)x (5/8)x - (3/5)x = 2 • Find the LCM for 8 and 5, which is 40: (25x/40) - (24x/40) = 2 (1x/40) = 2 • Solve for x: x = 2 × 40 = 80 Exam Strategy & Shortcut: Use the Denominator Divisibility Check. The total capacity must be a common multiple of the given denominators, 8 and 5. Their LCM is 40. Looking at the options (20, 30, 40, 80), both 40 and 80 are multiples of 40. You can quickly test 40: Initial = 25, Final = 24. Difference is 1 bottle (incorrect, we need a net difference of 2). Therefore, test 80: Initial = 50, Final = 48. Difference is 2 bottles. 80 is the correct answer. Common Pitfall: The most frequent mistake is setting up the equation as (5/8)x - 10 + 8x, mistakenly treating the 8 bottles added as a fraction of x rather than absolute units, or failing to realize that taking 10 and adding 8 simply means a net loss of 2. Final Answer: Therefore, the correct answer is 80.

Question 10

1 m of an object is to be represented by 10 cm on the drawing scale. The representative fraction (R.F.) of the scale is

Accepted Answer

Concept & Formula: This question tests the fundamental concept of Map Scaling and Representative Fraction (R.F.). The Representative Fraction is the ratio of the distance on the drawing (or map) to the actual distance on the ground. The critical rule is that both quantities must be in the same units before simplifying the fraction. R.F. = \frac{Distance on Drawing}{Actual Distance} Step-by-Step Solution: • Given: Distance on drawing = 10 cm Actual distance = 1 m • Calculation: Convert the actual distance into the same units as the drawing scale (centimeters): 1 m = 100 cm • Apply the R.F. formula: R.F. = \frac{10 cm}{100 cm} • Simplify the fraction by dividing numerator and denominator by 10: R.F. = (1/10) Exam Strategy & Shortcut: For linear scale questions, strictly focus on unit conversion. Memorize the metric ladder: 1 m = 100 cm. Plug the numbers in as \frac{Small Value}{Large Value} (since maps shrink reality) and ensure units match: (10/100) = (1/10). This should take less than 10 seconds. Common Pitfall: The single most common mistake is ignoring unit conversion and directly dividing the given numbers, leading to an R.F. of (10/1) = 10, or writing it blindly as (1/100) because they see "1 m". Always harmonize the units first. Final Answer: Therefore, the correct answer is (1/10).

Question 11

The fluid contained in a bucket can fill four large bottles or seven small bottles. A full large bottle is used to fill an empty small bottle. What fraction of the fluid is left over in the large bottle when the small one is full?

Accepted Answer

Concept & Logic: This is a comparative volume problem solvable through algebraic substitution or the LCM ratio method. By equating the total volume of the bucket in terms of both large and small bottles, we can establish a direct volume relationship (ratio) between one large bottle and one small bottle. Step-by-Step Solution: • Given: 1 Bucket = 4 Large Bottles (L) 1 Bucket = 7 Small Bottles (S) Therefore, 4L = 7S. • Calculation: Express the volume of a Large bottle in terms of a Small bottle: L = (7/4)S • A full large bottle (L) is poured into an empty small bottle until the small bottle is full (volume S removed). Find the remaining volume in the large bottle: Remaining Volume = L - S Remaining Volume = (7/4)S - S = (3/4)S • Calculate what fraction this remaining fluid is relative to the original large bottle's capacity (L): Fraction Left = \frac{Remaining Volume}{Original Volume L} Fraction Left = (\frac{3/4)S}{(7/4)S} Fraction Left = (3/4) × (4/7) = (3/7) Exam Strategy & Shortcut: Use the LCM Value Assumption Method. Let the capacity of the bucket be the LCM of 4 and 7, which is 28 units. Capacity of 1 Large bottle = (28/4) = 7 units. Capacity of 1 Small bottle = (28/7) = 4 units. If you pour from the Large (7 units) to fill the Small (4 units), you are left with 7 - 4 = 3 units in the Large bottle. The fraction left over in the large bottle is simply \frac{Leftover}{Total Large} = (3/7). This avoids fractions entirely! Common Pitfall: A common misstep is calculating the fraction relative to the small bottle or the whole bucket, rather than the initial volume of the *large bottle*. The question asks: "What fraction of the fluid is left over *in the large bottle*". The denominator must be the large bottle's capacity. Final Answer: Therefore, the correct answer is (3/7).

Question 12

To fill a tank, 25 buckets of water is required. How many buckets of water will be required to fill the same tank if the capacity of the bucket is reduced to two-fifth of its present?

Accepted Answer

Concept & Formula: This problem is governed by the principle of Inverse Proportion. When the total volume required to fill a container remains constant, the number of buckets needed is inversely proportional to the capacity of each bucket. Total Volume = Number of Buckets × Capacity per Bucket N_1 × C_1 = N_2 × C_2 Step-by-Step Solution: • Given: Initial number of buckets (N_1) = 25 Let initial bucket capacity be C_1 = x. New bucket capacity (C_2) = (2/5) of present = (2/5)x. • Calculation: Establish the total capacity of the tank using the initial conditions: Tank Volume = 25 × x = 25x • Set up the equation for the new conditions to find the new number of buckets (N_2): N_2 × ((2/5)x) = 25x • Solve for N_2 by dividing both sides by x and multiplying by the reciprocal of (2/5): N_2 = (25x/\frac{2){5}x} N_2 = 25 × (5/2) N_2 = (125/2) N_2 = 62.5 or 62(1/2) Exam Strategy & Shortcut: Use the Fractional Inverse Shortcut. If the capacity of a tool becomes a certain fraction ((a/b)) of its original size, the number of tools required to do the identical job becomes the reciprocal fraction ((b/a)) of the original number. Here, capacity becomes (2/5). Therefore, the number of buckets needed becomes (5/2) times the original amount. 25 × (5/2) = 62.5 This mental math maneuver allows you to solve the problem in seconds. Common Pitfall: A standard error is directly multiplying the original number of buckets by the new capacity fraction (25 × (2/5) = 10). This mistakenly assumes direct proportionality. Remember: smaller buckets mean *more* buckets are required, not fewer. Final Answer: Therefore, the correct answer is 62(1/2).

Question 13

A stationary engine has enough fuel to run 12 hours when its tank is $\frac{4}{5}$ full. How long will it run when the tank is $\frac{1}{3}$ full?

Accepted Answer

Concept & Logic: This problem is based on direct proportionality. The runtime of the engine is directly proportional to the amount of fuel in the tank. If we establish the total runtime for a completely full tank (1 whole unit), we can easily calculate the runtime for any other fractional capacity. Step-by-Step Solution: • Given: Runtime for (4/5) of the tank = 12 hours. • Calculation: First, find the runtime for a completely full tank. Let total runtime be T. (4/5) × T = 12 T = 12 × (5/4) T = 3 × 5 = 15 hours • Now, calculate the runtime when the tank is (1/3) full. New Runtime = (1/3) × T New Runtime = (1/3) × 15 New Runtime = 5 hours Exam Strategy & Shortcut: Use the Unitary Method Shortcut. If 4 parts out of 5 give 12 hours, then 1 part gives 12 \div 4 = 3 hours. The whole tank (5 parts) gives 5 × 3 = 15 hours. A tank that is (1/3) full will run for (1/3) of the total 15 hours, which is 5 hours. This mental path completely bypasses formal algebraic setup. Common Pitfall: A common mistake is inverting the fraction improperly when solving for the total time (e.g., multiplying 12 by (4/5) instead of (5/4)), which incorrectly reduces the total full-tank runtime to 9.6 hours. Always logically verify: a full tank MUST run longer than a partially full one. Final Answer: Therefore, the correct answer is 5 hours.

Question 14

A tree grows only $\frac{3}{5}$ as fast as the one beside it. In four years the combined growth of the trees is eight feet. How much does the shorter tree grow in 2 years?

Accepted Answer

Concept & Logic: This problem tests relative rates and linear equations. By defining the growth rate of the faster tree, we can express the growth rate of the slower tree as a fraction of it. Setting up an equation for their combined growth over a specific time period allows us to find their individual annual growth rates. Step-by-Step Solution: • Given: Combined growth in 4 years = 8 feet. Shorter tree's rate is (3/5) of the taller tree's rate. • Calculation: Let the annual growth rate of the taller tree be x feet/year. The annual growth rate of the shorter tree is (3/5)x feet/year. • Form an equation for their combined growth over 4 years: 4x + 4((3/5)x) = 8 x + (3/5)x = 2 (Dividing the entire equation by 4) (8/5)x = 2 x = 2 × (5/8) = (10/8) = 1.25 feet/year • Find the annual growth rate of the shorter tree: Shorter tree rate = (3/5) × 1.25 = 0.75 feet/year • Calculate the shorter tree's growth in 2 years: Growth = 2 × 0.75 = 1.5 feet • Compare 1.5 feet to the given options. 1.5 feet falls into the category of "Less than 2 feet". Exam Strategy & Shortcut: Use the Ratio Allocation Method. The growth ratio of shorter to taller is 3 : 5. Total combined growth in 4 years is 8 feet, meaning combined annual growth is 8 \div 4 = 2 feet/year. Divide this 2 feet/year into the 3 : 5 ratio (total 8 parts). 1 part = (2/8) = 0.25 feet. Shorter tree annual growth (3 parts) = 3 × 0.25 = 0.75 feet/year. In 2 years, it grows 2 × 0.75 = 1.5 feet. Common Pitfall: Students often solve for the taller tree's growth or forget to multiply the final annual rate by the required 2 years, leading them to search the options for 0.75 or 1.25, and guessing incorrectly when they don't find an exact match. Final Answer: Therefore, the correct answer is Less than 2 feet.

Question 15

The lowest temperature in the night in a city is one-third more than $\frac{1}{2}$ the highest during the day. Sum of the lowest temperature and the highest temperature is 100 degrees. Then what is the lowest temperature?

Accepted Answer

Concept & Logic: This algebraic word problem requires translating complex chained phrasing into a single equation. The phrase "one-third more than" applied to a base value means multiplying that base value by (1 + (1/3)), or (4/3). Careful translation of the conditions yields a system of two variables that easily collapses into one. Step-by-Step Solution: • Given: Sum of lowest (L) and highest (H) temperature = 100. L + H = 100 • Calculation: Translate the relationship for the lowest temperature: "one-third more than (1/2) the highest" Base value = (1/2)H Lowest (L) = Base + (1/3)(Base) L = (1/2)H + (1/3)((1/2)H) L = (1/2)H + (1/6)H L = (3/6)H + (1/6)H = (4/6)H = (2/3)H • Substitute L into the sum equation: (2/3)H + H = 100 (5/3)H = 100 H = 100 × (3/5) = 60 • Find the lowest temperature L: L = 100 - H = 100 - 60 = 40 degrees Exam Strategy & Shortcut: Use the Options Elimination Method combined with the L = (2/3)H ratio. If L = (2/3)H, then the ratio of L : H is 2 : 3. The total parts are 2 + 3 = 5 parts. Since the sum is 100, 1 part = 20. The lowest temperature is 2 parts, so 2 × 20 = 40. Checking the options immediately yields 40. Common Pitfall: The most frequent error is miswriting the equation for L as L = (1/3) + (1/2)H (adding one-third of a degree) rather than taking one-third *of* the half-value. Precise reading of "one-third more than [value]" is critical. Final Answer: Therefore, the correct answer is 40 degrees.

Question 16

Peter gave one-fourth of the amount he had to Michael. Michael in turn gave half of what he received from Peter to Sam. If the difference between the remaining amount with Peter and the amount received by Sam is ₹ 500, how much money did Michael rec…

Accepted Answer

Concept & Logic: This problem tracks the distribution of an initial amount through a series of fractional transfers. The strategy is to define Peter's initial money as a variable, track the exact fraction held by each person at the end of the transactions, and use the given absolute difference (₹ 500) to solve for the base amount. Step-by-Step Solution: • Given: Let Peter's initial amount be x. Peter gives (1/4)x to Michael. • Calculation: Track the amounts held by everyone: Peter's remaining amount = x - (1/4)x = (3/4)x Amount Michael receives = (1/4)x Amount Sam receives = (1/2) of Michael's amount = (1/2) × (1/4)x = (1/8)x • Apply the final condition: The difference between Peter's remaining amount and Sam's amount is 500. (3/4)x - (1/8)x = 500 • Find a common denominator to subtract the fractions: (6x/8) - (1x/8) = 500 (5x/8) = 500 x = 500 × (8/5) = 800 So, Peter's initial amount was ₹ 800. • The question asks for the amount Michael received from Peter: Michael's amount = (1/4)x = (1/4)(800) = 200 Exam Strategy & Shortcut: Use the LCM Assumption Method to avoid working with fractions. Look at the denominators in the problem: one-fourth (4) and half (2). We also know Sam gets half of a fourth, which is an eighth (8). Assume Peter's starting amount is 8 units. Peter gives Michael 2 units. Peter has 8 - 2 = 6 units left. Michael gives Sam 1 unit. Sam has 1 unit. Difference between Peter's remaining (6) and Sam's (1) is 5 units. We are told this difference is ₹ 500. So, 5 units = 500 \implies 1 unit = 100. Michael received 2 units from Peter. 2 × 100 = 200. Common Pitfall: Many students successfully calculate x = 800 (Peter's initial amount) and mistakenly choose 800 if it were an option, or get confused and mark "None of these" if it isn't. Always highlight the exact entity the question is asking for before selecting an option (in this case, what Michael received). Final Answer: Therefore, the correct answer is ₹ 200.

Question 17

Four children $A$, $B$, $C$ and $D$ divide a bag of sweets.  $A$ takes $\frac{1}{3}$ of them, $B$ $\frac{2}{5}$th of the remainder and the rest is equally shared between $C$ and $D$.  What fraction of the sweets did $C$ or $D$ get?

Accepted Answer

Concept & Logic: This problem is solved by tracking the remaining portion of a whole after sequential fractional deductions. When a whole (1) is reduced by a fraction, you multiply the remaining portion by the next fraction to find subsequent shares. Finally, dividing the ultimate remainder among a set number of people gives the individual share. Step-by-Step Solution: • Given: Total fraction of sweets = 1. A takes (1/3) of the total. • Calculation: Find the remainder after A takes their share: Remainder = 1 - (1/3) = (2/3) • Calculate B's share, which is (2/5) of the remainder: B's share = (2/5) × (2/3) = (4/15) • Find the new remainder after B takes their share: New Remainder = (2/3) - (4/15) New Remainder = (10/15) - (4/15) = (6/15) = (2/5) • This remaining (2/5) is shared equally between C and D. Divide it by 2 to find the individual share for C (or D): Share of C = (2/5) \div 2 = (2/5) × (1/2) = (1/5) Exam Strategy & Shortcut: Use the LCM Assumption Method. The denominators in the problem are 3 and 5. Their LCM is 15. Assume the total number of sweets in the bag is 15. A takes (1/3) of 15 = 5 sweets. Remaining sweets = 10. B takes (2/5) of 10 = 4 sweets. Remaining sweets = 10 - 4 = 6 sweets. C and D share the 6 sweets equally, so each gets 3 sweets. The fraction C got of the total is (3/15), which simplifies to (1/5). This avoids messy fraction arithmetic! Common Pitfall: A very common mistake is calculating B's share as (2/5) of the *total* instead of (2/5) of the *remainder*. Always pay close attention to whether fractions apply to the original amount or the leftover amount. Final Answer: Therefore, the correct answer is (1/5).

Question 18

A man spends one-quarter of his income on food, one-fifth of it on rent and remaining, which is ₹ 231, on other commodities.  Find his total income.

Accepted Answer

Concept & Formula: This is a classic basic fractions problem involving expenditure. Since both the food and rent expenses are expressed as fractions of the *total* income (not the remainder), we can simply add these fractions together to find the total fractional portion spent. The remainder can then be equated to the absolute monetary value given to find the total income. Step-by-Step Solution: • Given: Food expense = (1/4) of total. Rent expense = (1/5) of total. Remaining balance = ₹ 231. Let total income be x. • Calculation: Find the total fraction of income spent: Total Spent Fraction = (1/4) + (1/5) Total Spent Fraction = (5/20) + (4/20) = (9/20) • Find the fraction representing the remaining income: Remaining Fraction = 1 - (9/20) = (11/20) • Equate this remaining fraction to the actual remaining amount to solve for x: (11/20)x = 231 x = 231 × (20/11) x = 21 × 20 = 420 Exam Strategy & Shortcut: Use the LCM Value Assumption. The denominators are 4 and 5, so assume the total income is a multiple of 20 units. Food = (1/4) of 20 = 5 units. Rent = (1/5) of 20 = 4 units. Total spent = 9 units. Remaining = 20 - 9 = 11 units. We are told the remaining amount is 231. Thus, 11 units = 231, meaning 1 unit = 21. Total income is 20 units, so 20 × 21 = 420. Common Pitfall: The main trap is treating the "one-fifth" rent expense as a fraction of the *remaining* income after food, rather than a fraction of the *total* income. The phrasing "one-fifth of it" refers back to the original subject, "his income." Final Answer: Therefore, the correct answer is ₹ 420.

Question 19

A boy read $\frac{3}{8}$th of a book on one day and $\frac{4}{5}$th of the remainder on another day.  If there were 30 pages unread, how many pages did the book contain?

Accepted Answer

Concept & Logic: This problem utilizes sequential fraction reduction. Unlike standard fraction addition, the second fraction applies only to the "remainder" of the first operation. You must calculate the remaining fraction after the first day, apply the second fraction to that specific remainder, and then equate the final unread fraction to the absolute number of pages left. Step-by-Step Solution: • Given: First day reading = (3/8) of total. Second day reading = (4/5) of the remainder. Unread pages = 30. Let total pages be x. • Calculation: Determine the fraction of the book remaining after the first day: Remainder after Day 1 = 1 - (3/8) = (5/8) • Calculate the fraction of the *total* book read on the second day: Read on Day 2 = (4/5) × (5/8) = (4/8) = (1/2) • Determine the total fraction of the book read over both days: Total Read = (3/8) + (1/2) = (3/8) + (4/8) = (7/8) • Find the fraction representing the unread portion: Unread Fraction = 1 - (7/8) = (1/8) • Equate the unread fraction to the given number of unread pages: (1/8)x = 30 x = 30 × 8 = 240 Exam Strategy & Shortcut: Use the Successive Remaining Fraction Multiplier. If he reads (3/8), the unread portion is (5/8). If he then reads (4/5) of *that*, the new unread portion is (1/5) of it. So, the final unread fraction of the whole book is simply the product of the unread ratios: (5/8) × (1/5) = (1/8). Since (1/8) of the book is 30 pages, the whole book is 30 × 8 = 240 pages. This single-line multiplication takes only seconds. Common Pitfall: A very common mistake is simply adding (3/8) and (4/5) together, completely ignoring the crucial phrase "of the remainder". This leads to an impossible fraction larger than 1. Final Answer: Therefore, the correct answer is 240.

Question 20

An institute organised a fete and $\frac{1}{5}$ of the girls and $\frac{1}{8}$ of the boys participated in the same.  What fraction of the total number of students took part in the fete?

Accepted Answer

Concept & Logic: This problem introduces the concept of weighted averages and Data Sufficiency in algebra. To find a combined fraction for a total population, you must know the ratio or absolute counts of the individual subgroups. The overall participation fraction is a weighted average of (1/5) and (1/8), meaning its exact value depends entirely on how many boys versus girls there are in the institute. Step-by-Step Solution: • Given: Fraction of girls participating = (1/5). Fraction of boys participating = (1/8). • Calculation: Let the total number of girls be G and the total number of boys be B. Number of participating girls = (1/5)G Number of participating boys = (1/8)B • Set up the expression for the total fraction of participating students: Total Fraction = \frac{Participating Girls + Participating Boys}{Total Girls + Total Boys} Total Fraction = (\frac{1/5)G + (1/8)B}{G + B} • Because we do not have an equation linking G and B (i.e., we don't know the ratio of boys to girls), we cannot simplify this expression into a single numerical fraction. The value will shift depending on the gender ratio of the school. Exam Strategy & Shortcut: Recognize the Missing Weight Condition immediately. You have rates ((1/5) and (1/8)) but no absolute values or relative weights (ratios) connecting the two different groups (Boys and Girls). Without knowing the ratio of Boys to Girls, it is mathematically impossible to find the combined average rate. Mark "Data inadequate" and move on within 5 seconds. Common Pitfall: A standard trap is assuming there is an equal number of boys and girls (G = B), which would lead you to calculate a simple average: (\frac{1/5) + (1/8)}{2} = (13/80), and then likely guess an incorrect option. Never assume equal weights unless explicitly stated. Final Answer: Therefore, the correct answer is Data inadequate.

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