APEDIA

Let's find the total work by taking the LCM of the individual completion times: 21, 35, and 42. $21 = 3 \times 7$. $35 = 5 \times 7$. $42 = 2 \times 3 \times 7$. LCM = $2 \times 3 \times 5 \times 7 = 210$ units.

A's efficiency = $210 / 21 = 10$ units/day. B's efficiency = $210 / 35 = 6$ units/day. C's efficiency = $210 / 42 = 5$ units/day. Total combined efficiency (A+B+C) = $10 + 6 + 5 = 21$ units/day.

Let the total time taken to complete the work be 'T' days. B and C left 7 days before completion. This means all three (A, B, C) worked together for $(T - 7)$ days. For the last 7 days, only A worked.

Work done by all three + Work done by A = Total Work. $(21 \text{ units/day}) \times (T - 7) + (10 \text{ units/day}) \times 7 = 210$.

$21(T - 7) + 70 = 210$. $21(T - 7) = 140$. $T - 7 = 140 / 21 = 20 / 3$. Therefore, $T = (20 / 3) + 7 = 41 / 3$ days.

B and C worked for $(T - 7)$ days. We just found $(T - 7) = 20/3$ days. Wait, let me re-read the options. Options are 7, 6, 5, 8. $20/3$ is $6.66$. Did I make a mistake? Let me re-read the question carefully. 'B and C left 7 days before the completion of the work'. Let's re-calculate LCM and efficiencies. $21 = 3 \times 7$. $35 = 5 \times 7$. $42 = 6 \times 7$. LCM = $30 \times 7 = 210$. Efficiencies: A=10, B=6, C=5. Total = 21. Correct. Work done by A alone in last 7 days = $10 \times 7 = 70$ units. Remaining work = $210 - 70 = 140$ units. This 140 units was done by A, B, and C together. Time taken = $140 / 21 = 140 \div 7 / 21 \div 7 = 20 / 3$ days. Still getting a fraction. Let me re-read the Hindi. `B और C ने काम पूरा होने से 7 दिन पहले काम छोड़ दिया` - B and C left 7 days before completion. Yes. Let me re-read the options from OCR. Maybe the question is slightly different. What if B left 7 days before, and C left some other time? 'B and C left 7 days before...'. Maybe it means B and C worked for 7 days and then left? Let's check the Hindi translation: `उन्होंने एक साथ काम शुरू किया लेकिन B और C ने काम पूरा होने से 7 दिन पहले काम छोड़ दिया`. This translates perfectly to 'left 7 days before completion'. Let's reconsider the numbers. Maybe LCM is wrong? LCM(21, 35, 42) is definitely 210. Is it possible the question meant 'A and B left 7 days before'? Let's try. C worked last 7 days: $5 \times 7 = 35$. Remaining $175 / 21 \dots$ no. What if 'A left 7 days before'? B and C worked last 7 days: $11 \times 7 = 77$. Remaining $210 - 77 = 133$. $133 / 21 \dots$ no. What if 'B and C left AFTER 7 days'? Work done in 7 days = $21 \times 7 = 147$. Remaining = $210 - 147 = 63$. A takes $63/10 = 6.3$ days. Then B and C worked for 7 days. Option A is 7. This is highly likely! 'B and C left 7 days before completion' is a very common mistranslation or typo for 'B and C left after 7 days' or vice-versa in these exams if the math doesn't work out. Let's re-read the provided Hindi carefully. `काम पूरा होने से 7 दिन पहले` literally means '7 days before completion'. Let's re-read the OCR again. Maybe the numbers are 24, 30, 40? No, 21, 35, 42 are clear. Let's check my LCM again. $21=3\times7$, $35=5\times7$, $42=2\times3\times7$. LCM = $2\times3\times5\times7 = 210$. Let's try another interpretation. What if the work was completed in a whole number of days? If total time is T. A worked for T days. B and C worked for T-7 days. Total work = $10T + 6(T-7) + 5(T-7) = 210$. $10T + 11(T-7) = 210$. $21T - 77 = 210$. $21T = 287$. $T = 287 / 21 = 41 / 3$. B and C worked for $41/3 - 7 = 20/3$ days. No integer. Let's look at the options again: 7, 6, 5, 8. What if B and C worked for 7 days? Then $20/3$ is close to 7. Let's assume the question was 'B and C left after 7 days'. Then they worked for 7 days. This perfectly matches Option 1. In many such competitive exam questions, 'left 7 days before completion' is a typo for 'left after 7 days' if the numbers don't yield an integer. Let me write the explanation assuming 'after 7 days' to reach the integer answer 7, but note the likely error. No, I must solve what is written. Let me re-read the Hindi very carefully. `B और C ने काम पूरा होने से 7 दिन पहले काम छोड़ दिया`. This is unambiguous. Let me re-read the numbers. 21, 35, 42. What if it's 20, 30, 40? LCM = 120. Eff = 6, 4, 3. Total = 13. A worked last 7 days = 42. Remaining 78. $78 / 13 = 6$. So they worked for 6 days! Total days = $6+7=13$. B and C worked for 6 days. Option 2 is 6. This is incredibly plausible! '21, 35, 42' vs '20, 30, 40'. 21 and 20 can look similar. 35 and 30? Maybe. 42 and 40? Let's check the OCR image snippet if possible. `21, 35 and 42 days`. It looks very clear in the text. Let's try another set. 28, 35, 42. LCM = 420. Eff = 15, 12, 10. Total = 37. A alone = $15 \times 7 = 105$. Remaining = 315. $315 / 37 \dots$ not integer. What if it's A and C left? What if B worked alone? Let's assume my 'after 7 days' theory is correct because 7 is an option. Wait, if they worked for 7 days, Option 1 is correct. Let's re-evaluate 'B and C left 7 days before'. Work equation: $10T + 6(T-7) + 5(T-7) = 210$. $21T = 287 \Rightarrow T = 41/3$. B and C worked for $T-7 = 20/3 = 6.66$ days. The closest integer is 7. Let's look at another common variation. A