The question gives the codes ROPE = 5391 and OILY = 2458 and asks for the code of the letter O. In such questions, the letters and digits are coded without a fixed order, so we need to find the letter common to both words and the digit common to both codes.
Identify the given information:
Find the common letter:
ROPE has the letters R, O, P, E and OILY has the letters O, I, L, Y. The only letter in both words is O.
Find the common digit:
The code 5391 has the digits 5, 3, 9, 1 and the code 2458 has the digits 2, 4, 5, 8. The only digit in both codes is 5.
Match the common letter with the common digit:
Since O is the only common letter and 5 is the only common digit, O is coded as 5.
Correct Option: B) 5 — O is the only letter shared by ROPE and OILY, and 5 is the only digit shared by 5391 and 2458. Hence, Option B is the correct answer.
A) 8: This is incorrect because 8 appears only in the code for OILY, so it belongs to I, L or Y.
C) 3: This is incorrect because 3 appears only in the code for ROPE, so it belongs to R, P or E.
D) 2: This is incorrect because 2 appears only in the code for OILY, so it belongs to I, L or Y.
To find the number of diagonals in a regular octagon, we can use the general formula for the number of diagonals in a polygon with \(n\) sides. An octagon has 8 sides, so we will substitute \(n=8\) into the formula.
Thus, a regular octagon has 20 diagonals.
A) 20 is the correct number of diagonals for a regular octagon, as calculated using the formula.
The problem requires us to find the missing letter-clusters (# and %) by identifying the pattern or relationship between the given letter-cluster pairs. We need to analyze the transformation from the first cluster to the second cluster in the known pair (DAF : %) and apply the reverse transformation to find # from LIN, or apply the same transformation to find % from DAF.
Let's analyze the relationship between the letters in the given options and the known cluster LIN. We need to find a cluster # such that the transformation from # to LIN follows a consistent rule.
Let's examine Option C: # = OLQ, % = AXC.
Now, let's apply this pattern to the second pair: DAF : %.
Since both pairs follow the same pattern (subtract 3 from each letter's alphabetical position), Option C is the correct choice.
C) # = OLQ, % = AXC
As shown in the step-by-step analysis, if we take OLQ and subtract 3 from the alphabetical position of each letter, we get LIN (O-3=L, L-3=I, Q-3=N). Similarly, if we take DAF and subtract 3 from the alphabetical position of each letter, we get AXC (D-3=A, A-3=X, F-3=C).
To find a single equivalent discount for consecutive discounts, we can assume an initial price (e.g., 100 units) and apply each discount sequentially. Alternatively, we can use the formula for equivalent discount for two successive discounts and extend it, or directly calculate the final price after all discounts and then find the total discount percentage.
Assume an initial price: Let the original price of an item be \(P = 100\).
Apply the first discount (13%): After a 13% discount, the price becomes: \[ P_1 = P \times \left(1 - \frac{13}{100}\right) = 100 \times (1 - 0.13) = 100 \times 0.87 = 87 \] So, the price after the first discount is 87.
Apply the second discount (20%): This discount is applied to the new price \(P_1\). The price becomes: \[ P_2 = P_1 \times \left(1 - \frac{20}{100}\right) = 87 \times (1 - 0.20) = 87 \times 0.80 = 69.60 \] So, the price after the second discount is 69.60.
Apply the third discount (4%): This discount is applied to the latest price \(P_2\). The final price becomes: \[ P_3 = P_2 \times \left(1 - \frac{4}{100}\right) = 69.60 \times (1 - 0.04) = 69.60 \times 0.96 = 66.816 \] So, the final price after all three discounts is 66.816.
Calculate the total discount: The total discount is the difference between the original price and the final price: \[ \text{Total Discount Amount} = P - P_3 = 100 - 66.816 = 33.184 \]
Calculate the single equivalent discount percentage: Since we started with an original price of 100, the total discount amount directly represents the percentage discount: \[ \text{Equivalent Discount Percentage} = \frac{\text{Total Discount Amount}}{\text{Original Price}} \times 100 = \frac{33.184}{100} \times 100 = 33.184\% \]
Round up to two decimal places: Rounding 33.184% to two decimal places gives 33.18%.
C) 33.18%
Seven people A, B, C, D, R, S and T sit around a circular table facing the centre, and several positional clues are given. We need to fix the arrangement using the clues and then count the people between D and R when moving from the right of D.
Identify the given information:
Recall the direction rule:
When people sit around a table facing the centre, moving to a person's left means moving clockwise, and moving to a person's right means moving anticlockwise.
Place T, B, A and R:
Number the seats clockwise from 1 and let T be at seat 7. B is third to the left of T, so B is three seats clockwise, at seat 3. Only T sits between A and R, so A and R occupy seats 6 and 1.
Place S, C and D:
R is second to the left of S, so R is two seats clockwise from S. If R were at seat 1, S would be at seat 6, which would then belong to A. So, R is at seat 6, S is at seat 4 and A is at seat 1. The free seats are 2 and 5; C cannot sit at seat 2 (next to B), so C is at seat 5 and D is at seat 2.
So, the final clockwise arrangement is: A (1), D (2), B (3), S (4), C (5), R (6), T (7).
Count from the right of D:
Moving to the right of D means moving anticlockwise from seat 2. The people met before reaching R are A and T.
\[ \text{People between D and R} = 2 \]Correct Option: D) Two — Moving anticlockwise from D, the people A and T sit before R. Hence, Option D is the correct answer.
A) Three: This is incorrect because three people (B, S and C) sit between D and R only when counted from the left of D.
B) Four: This is incorrect because only two people sit between D and R from the right of D.
C) One: This is incorrect because two people (A and T) sit between D and R from the right of D.
The question asks for the name of the new legislation enacted by the Government of India in 2024 to replace the Indian Telegraph Act, 1885, and modernize the telecom sector. This requires knowledge of recent legislative developments in India's telecommunications policy.
B) The Telecommunication Act, 2023 is the correct answer. This Act was passed by the Parliament in December 2023 and subsequently notified to replace the Indian Telegraph Act, 1885, the Indian Wireless Telegraphy Act, 1933, and the Telegraph Wires (Unlawful Possession) Act, 1950, thereby modernizing the telecom sector's legal framework.
To solve this problem, we need to determine the combined filling/emptying rate of all three taps. We'll calculate the work done by each tap in one hour and then combine these rates. Finally, we'll use the combined rate to find the time required to fill one-fourth of the cistern.
Calculate the filling rate of the first tap:
The first tap fills the cistern in 4 hours. So, in one hour, it fills \(\frac{1}{4}\) of the cistern.
\[ \text{Rate of Tap 1} = \frac{1}{4} \text{ cistern/hour} \]Calculate the filling rate of the second tap:
The second tap fills the cistern in 32 hours. So, in one hour, it fills \(\frac{1}{32}\) of the cistern.
\[ \text{Rate of Tap 2} = \frac{1}{32} \text{ cistern/hour} \]Calculate the emptying rate of the third tap:
The third tap empties the cistern in 4 hours. So, in one hour, it empties \(\frac{1}{4}\) of the cistern. We represent emptying as a negative rate.
\[ \text{Rate of Tap 3} = -\frac{1}{4} \text{ cistern/hour} \]Calculate the combined rate of all three taps:
When all three taps are opened together, their combined rate is the sum of their individual rates:
\[ \text{Combined Rate} = \text{Rate of Tap 1} + \text{Rate of Tap 2} + \text{Rate of Tap 3} \] \[ \text{Combined Rate} = \frac{1}{4} + \frac{1}{32} - \frac{1}{4} \]Notice that the \(\frac{1}{4}\) (filling) and \(-\frac{1}{4}\) (emptying) terms cancel each other out.
\[ \text{Combined Rate} = \frac{1}{32} \text{ cistern/hour} \]This means that with all three taps open, the cistern fills at a rate of \(\frac{1}{32}\) of the cistern per hour.
Determine the fraction of the cistern to be filled:
We need to fill one-fourth of the empty cistern.
\[ \text{Fraction to fill} = \frac{1}{4} \]Calculate the time required to fill one-fourth of the cistern:
Time is calculated as the amount of work (fraction to fill) divided by the rate.
\[ \text{Time} = \frac{\text{Fraction to fill}}{\text{Combined Rate}} \] \[ \text{Time} = \frac{\frac{1}{4}}{\frac{1}{32}} \] \[ \text{Time} = \frac{1}{4} \times 32 \] \[ \text{Time} = 8 \text{ hours} \]C) 8
The calculation shows that it will take 8 hours to fill one-fourth of the cistern when all three taps are opened together.
The problem requires us to modify a given number based on specific rules for its odd and even digits, then identify two specific digits in the new number, and finally calculate their sum.
Let's process each digit:
The sum of the digits which are second from the left and second from the right in the new number is 10.
B) 10 is the correct sum calculated from the modified digits.
The question gives four letter-cluster pairs, three of which follow the same rule, and asks for the pair that does not belong. We need to compare each letter of the first cluster with the corresponding letter of the second cluster using alphabet positions and find the pair that breaks the common rule.
Identify the given information:
Find the rule using BU – GK:
The first letter moves 5 places forward and the second letter moves 10 places back.
\[ B(2)+5=7=G \] \[ U(21)-10=11=K \]Check YR – DH and MF – RV:
Counting continues around the alphabet after Z and before A.
\[ Y(25)+5=30-26=4=D,\quad R(18)-10=8=H \] \[ M(13)+5=18=R,\quad F(6)-10=-4+26=22=V \]Both pairs follow the rule.
Check TN – YC:
\[ T(20)+5=25=Y \] \[ N(14)-10=4=D \]The rule gives TN – YD, but the pair shows TN – YC. So, this pair does not follow the rule.
Correct Option: C) TN – YC — In the other pairs the first letter moves 5 places forward and the second moves 10 places back, but N − 10 gives D, not C. Hence, Option C is the correct answer.
A) BU – GK: This is incorrect because B + 5 = G and U − 10 = K, so it follows the rule.
B) YR – DH: This is incorrect because Y + 5 = D and R − 10 = H, so it follows the rule.
D) MF – RV: This is incorrect because M + 5 = R and F − 10 = V, so it follows the rule.
This problem involves substituting new meanings for mathematical operators and then evaluating the expression according to the order of operations (BODMAS/PEMDAS).
Therefore, the value of the expression is 173.
C) 173 is the correct answer, obtained by correctly substituting the operators and following the order of operations.