Questions: Algebra Play
Algebra Play24 min readLesson 6 of 6
Q1 Explain why the trick "double, add 4, halve, subtract the number" always gives 2.
Answer (English): Start with x. After doubling, adding 4 and halving, the expression is (2x + 4)/2 = x + 2. Subtracting x leaves 2. The letter cancels, so the answer is independent of the starting number. Changing the added number to 6 gives a final answer of 3.
Hinglish: x se shuru karo. Double, 4 jodne aur aadha karne par (2x + 4)/2 = x + 2 milta hai. x ghatane par 2 bachta hai. Letter cancel ho jata hai, isliye answer starting number par depend nahi karta. 6 jodo to answer 3 aata hai.
Q2 In the date trick, why does subtracting 165 recover the month and day?
Answer (English): The steps produce 100M + 165 + D, where M is the month and D is the day. Subtracting 165 leaves 100M + D. Since D is at most 31, it needs two digits, so the last two digits are the day and the remaining digits give the month. For example, 1269 − 165 = 1104 gives 4 November.
Hinglish: Steps se 100M + 165 + D banta hai. 165 ghatao to 100M + D bachta hai. Day maximum 31 hota hai, isliye aakhri do digits day aur baaki digits month hain. Jaise 1269 − 165 = 1104 se 4 November milta hai.
Q3 Find the top of a three-row pyramid with bottom row 5, 8 and 2.
Answer (English): The middle row is 5 + 8 = 13 and 8 + 2 = 10. The top is 13 + 10 = 23. The formula gives 5 + 2 × 8 + 2 = 23 as well, because the middle number is counted twice.
Hinglish: Middle row 5 + 8 = 13 aur 8 + 2 = 10 hai. Top 13 + 10 = 23. Formula se bhi 5 + 2 × 8 + 2 = 23 aata hai kyunki beech ka number do baar judta hai.
Q4 Explain why a four-row pyramid has coefficients 1, 3, 3, 1.
Answer (English): Label the bottom row a, b, c, d. The next row has a + b, b + c, c + d; the third row has a + 2b + c and b + 2c + d; the top adds them to a + 3b + 3c + d. The middle numbers are used three times each, while the ends are used once.
Hinglish: Bottom row a, b, c, d lo. Agla row a + b, b + c, c + d; teesra row a + 2b + c aur b + 2c + d; top unko jodta hai to a + 3b + 3c + d milta hai. Beech ke numbers teen-teen baar use hote hain aur ends ek baar.
Q5 A 2 × 2 calendar block has sum 44. Find its four numbers.
Answer (English): The sum is 4a + 16 = 44, so 4a = 28 and a = 7. The block is 7, 8, 14 and 15. Check: 7 + 8 + 14 + 15 = 44.
Hinglish: Sum 4a + 16 = 44, isliye 4a = 28 aur a = 7. Block 7, 8, 14, 15 hai. Check: 7 + 8 + 14 + 15 = 44.
Q6 Find the largest product using the digits 2, 4 and 6 once each in a two-digit number times a one-digit number.
Answer (English): The largest digit becomes the multiplier, and the other two digits in decreasing order form the two-digit number: 42 × 6 = 252. Compare: 42 × 6 = 252, 62 × 4 = 248, 24 × 6 = 144, 46 × 2 = 92, 26 × 4 = 104 and 64 × 2 = 128. The largest is 42 × 6 = 252.
Hinglish: Largest digit multiplier banta hai aur baaki do digits decreasing order mein milkar do-digit number banate hain: 42 × 6 = 252. Baaki candidates 42 × 6 = 252, 62 × 4 = 248, 24 × 6 = 144, 46 × 2 = 92, 26 × 4 = 104 hain. Sabse bada 42 × 6 = 252 hai.
Q7 Why is the difference between a two-digit number and its reversal always divisible by 9?
Answer (English): Write the number as 10a + b and its reversal as 10b + a. Subtracting gives 9b − 9a = 9(b − a), which is always a multiple of 9. The quotient is b − a, the difference between the digits.
Hinglish: Number 10a + b aur reversal 10b + a likho. Ghatane par 9b − 9a = 9(b − a) milta hai, jo hamesha 9 ka multiple hai. Quotient b − a hota hai, yaani digits ka difference.
Q8 Show that abcabc is divisible by 7, 11 and 13.
Answer (English): abcabc = 1001 × abc. Since 1001 = 7 × 11 × 13, the number has all three as factors. Dividing by 7, then 11, then 13 leaves abc.
Hinglish: abcabc = 1001 × abc. 1001 = 7 × 11 × 13 hai, isliye teeno factors hain. 7, phir 11, phir 13 se divide karo to abc bachta hai.
Q9 A farm has 40 heads and 110 legs. How many horses and how many hens?
Answer (English): Let horses be h and hens n. Then h + n = 40 and 4h + 2n = 110. Dividing the second equation by 2 gives 2h + n = 55. Subtracting the head equation gives h = 15, so n = 25.
Hinglish: Horses h aur hens n lo. h + n = 40 aur 4h + 2n = 110. Doosri equation ko 2 se divide karo: 2h + n = 55. Head equation ghatane par h = 15, isliye n = 25.
Q10 Karim starts with 12 coins, doubles each round and pays 8 to the genie. How many coins does he have after three rounds? Should he accept the deal?
Answer (English): After r rounds the coins are 2ʳ × 12 − 8(2ʳ − 1). For r = 3 this is 8 × 12 − 8 × 7 = 96 − 56 = 40. He should accept because the cost per round (8) is less than his starting number (12), so he ends with more than he began.
Hinglish: r rounds ke baad coins 2ʳ × 12 − 8(2ʳ − 1) hote hain. r = 3 par 8 × 12 − 8 × 7 = 96 − 56 = 40. Deal accept karni chahiye kyunki cost 8 shuruaati 12 se kam hai, isliye end mein zyada coins bachte hain.
Quiz pointer
The five-question quiz for this chapter is separate from these answers. Take it without looking back, then redo any missed question by writing the trick or pyramid steps with a letter-number. A quick check on a small example catches most slips.