💡First, I need to determine the total number of members in the sports club, which is 30.Next, I note the number of members who play badminton (17) and those who play tennis (21).Since 10 members play both badminton and tennis, I need to account for this overlap to avoid double-counting.Using the principle of inclusion-exclusion, I add the number of badminton players to the number of tennis players and then subtract the number of members who play both sports:17 (badminton) + 21 (tennis) - 10 (both) = 28.This result represents the total number of members who play at least one of the two sports.Finally, to find out how many members do not play either badminton or tennis, I subtract the number of members who play at least one sport from the total number of members:30 (total members) - 28 (at least one sport) = 2.Therefore, there are 2 members who do not play either badminton or tennis.

💡Okay, so I have this problem here: What is the 123rd digit after the decimal point when 47 divided by 740 is expressed as a decimal? Hmm, that sounds a bit tricky, but let me try to figure it out step by step.First, I know that to find the decimal representation of a fraction, I can perform long division. So, I need to divide 47 by 740. Let me set that up. 47 divided by 740. Since 47 is smaller than 740, I know the decimal will start with 0., and then I'll add zeros to 47 to continue the division.Let me write it out:47.000000...Divided by 740.So, 740 goes into 470 zero times. I write down 0. and then bring down a zero, making it 470. Still, 740 doesn't go into 470, so I write another 0, making it 4700.Now, how many times does 740 go into 4700? Let me calculate that. 740 times 6 is 4440, and 740 times 7 is 5180, which is too big. So, 6 times. I write down 6, subtract 4440 from 4700, which leaves me with 260.Bring down another zero, making it 2600. How many times does 740 go into 2600? Let's see, 740 times 3 is 2220, and 740 times 4 is 2960, which is too big. So, 3 times. I write down 3, subtract 2220 from 2600, leaving 380.Bring down another zero, making it 3800. 740 times 5 is 3700, and 740 times 6 is 4440, which is too big. So, 5 times. I write down 5, subtract 3700 from 3800, leaving 100.Bring down another zero, making it 1000. 740 goes into 1000 once, because 740 times 1 is 740. I write down 1, subtract 740 from 1000, leaving 260.Wait a minute, I've seen this remainder before. Earlier, after the first division, I had 260. That means the decimal is starting to repeat now. So, the repeating cycle is from here.Let me recap the decimal I've got so far: 0.06351...But wait, actually, let me check my calculations again because I think I might have made a mistake in the decimal places.Starting over:47 divided by 740. 740 goes into 470 zero times. So, 0.Bring down a zero: 4700.740 goes into 4700 six times (6*740=4440). Subtract 4440 from 4700: 260.Bring down a zero: 2600.740 goes into 2600 three times (3*740=2220). Subtract 2220 from 2600: 380.Bring down a zero: 3800.740 goes into 3800 five times (5*740=3700). Subtract 3700 from 3800: 100.Bring down a zero: 1000.740 goes into 1000 once (1*740=740). Subtract 740 from 1000: 260.Now, we're back to 260, which was the remainder after the first division. So, the decimal repeats from here.So, the decimal expansion is 0.0635189189189..., where "189" repeats indefinitely.Wait, let me write out the decimal places properly:After dividing, we have:- 0.0 (from 47/740)- Then, 6 (from 4700/740)- Then, 3 (from 2600/740)- Then, 5 (from 3800/740)- Then, 1 (from 1000/740)- Then, 8 (from 2600 again? Wait, no, after 1000, we had 260, then 2600, which was 3 times, but that was already counted.Wait, I think I messed up the decimal places. Let me try to write the decimal correctly.Starting from the beginning:47 divided by 740:- 47.000000...- 740 goes into 470 zero times. So, 0.- Bring down a zero: 4700.- 740 goes into 4700 six times (6*740=4440). Subtract: 4700-4440=260.- Bring down a zero: 2600.- 740 goes into 2600 three times (3*740=2220). Subtract: 2600-2220=380.- Bring down a zero: 3800.- 740 goes into 3800 five times (5*740=3700). Subtract: 3800-3700=100.- Bring down a zero: 1000.- 740 goes into 1000 once (1*740=740). Subtract: 1000-740=260.Now, we're back to 260, which was the remainder after the first division. So, the decimal repeats from here.So, the decimal expansion is 0.0635189189189..., where "189" repeats indefinitely.Wait, let me count the decimal places:- After the decimal point: 0 (tenths place), 6 (hundredths), 3 (thousandths), 5 (ten-thousandths), 1 (hundred-thousandths), 8 (millionths), 9 (ten-millionths), 1 (hundred-millionths), 8 (billionths), 9 (ten-billionths), and so on.So, the repeating cycle is "189", which is three digits long.But wait, looking back, after the decimal, we have 0.0635189189189...So, the non-repeating part is "0635", and then "189" repeats.Wait, no, actually, the repeating part starts after the decimal. Let me see:From the division, after the decimal, we have:0.0 (from the first division)Then, 6 (from 4700/740)Then, 3 (from 2600/740)Then, 5 (from 3800/740)Then, 1 (from 1000/740)Then, 8 (from 2600 again? Wait, no, after 1000, we had 260, then 2600, which was 3 times, but that was already counted.Wait, I think I'm getting confused. Let me try to write down the decimal places step by step:1. 47.000000... divided by 740.2. 740 goes into 470 zero times. So, 0.3. Bring down a zero: 4700.4. 740 goes into 4700 six times (6). Subtract 4440, remainder 260.5. Bring down a zero: 2600.6. 740 goes into 2600 three times (3). Subtract 2220, remainder 380.7. Bring down a zero: 3800.8. 740 goes into 3800 five times (5). Subtract 3700, remainder 100.9. Bring down a zero: 1000.10. 740 goes into 1000 once (1). Subtract 740, remainder 260.11. Bring down a zero: 2600. Now, we've seen this before.So, the decimal is 0.0635189189189...Wait, so after the decimal, the first digit is 0, then 6, then 3, then 5, then 1, then 8, then 9, and then it repeats 1, 8, 9, etc.So, the repeating cycle is "189", which is three digits long.But wait, the decimal starts with 0.0635189189189...So, the non-repeating part is "0635", and then "189" repeats.Wait, no, actually, the repeating part starts after the decimal. Let me see:From the division, after the decimal, we have:- 0 (from the first division)- 6 (from 4700/740)- 3 (from 2600/740)- 5 (from 3800/740)- 1 (from 1000/740)- 8 (from 2600 again? Wait, no, after 1000, we had 260, then 2600, which was 3 times, but that was already counted.Wait, I think I'm overcomplicating this. Let me just write out the decimal expansion properly.After performing the division, the decimal expansion of 47/740 is 0.0635189189189..., where "189" repeats indefinitely.So, the decimal is 0.0635189189189...Now, to find the 123rd digit after the decimal point, I need to determine where in the repeating cycle the 123rd digit falls.First, let's identify the repeating cycle. From the decimal expansion, the repeating part is "189", which is three digits long.But wait, looking at the decimal, it's 0.0635189189189...So, after the decimal, the digits are:1: 02: 63: 34: 55: 16: 87: 98: 19: 810: 911: 112: 813: 9... and so on.Wait, so the repeating cycle starts at the fifth digit, which is 1, and then repeats "189" every three digits.So, the non-repeating part is "0635", which is four digits, and then the repeating cycle "189" starts.Therefore, the decimal is structured as:0.0635 189 189 189...So, positions 1-4: 0,6,3,5Positions 5-7: 1,8,9Positions 8-10: 1,8,9And so on.So, to find the 123rd digit, we need to see how many digits are in the non-repeating part and then how many are in the repeating cycle.The non-repeating part is four digits: positions 1-4.Then, starting from position 5, the repeating cycle "189" begins, which is three digits long.So, the total number of digits after the decimal is:- 4 non-repeating digits- Then, repeating every 3 digits.So, to find the 123rd digit, we subtract the first 4 digits, and then see where the 123rd digit falls in the repeating cycle.So, 123 - 4 = 119 digits into the repeating cycle.Now, since the repeating cycle is 3 digits long, we can find the position within the cycle by dividing 119 by 3 and finding the remainder.119 divided by 3 is 39 with a remainder of 2.So, 39 full cycles (39*3=117 digits) and then 2 more digits into the next cycle.So, the 119th digit in the repeating cycle is the 2nd digit of the cycle "189".Looking at "189", the digits are:1 (position 1)8 (position 2)9 (position 3)So, the 2nd digit is 8.Therefore, the 123rd digit after the decimal point is 8.Wait, but let me double-check my steps to make sure I didn't make a mistake.1. Perform long division of 47 by 740 to get the decimal expansion.2. Identified that the decimal is 0.0635189189189..., with "189" repeating.3. Determined that the non-repeating part is four digits: 0,6,3,5.4. The repeating cycle "189" starts at the fifth digit.5. To find the 123rd digit, subtract the first four digits: 123 - 4 = 119.6. Since the cycle is three digits, divide 119 by 3: 119 ÷ 3 = 39 with a remainder of 2.7. The remainder 2 indicates the second digit in the cycle "189", which is 8.Therefore, the 123rd digit after the decimal is 8.I think that's correct. But just to be thorough, let me try to write out the decimal expansion up to, say, the 15th digit to see the pattern.Decimal expansion: 0.0635189189189...Positions:1: 02: 63: 34: 55: 16: 87: 98: 19: 810: 911: 112: 813: 914: 115: 8So, yes, the pattern is clear. After the first four digits, "189" repeats every three digits.So, for the 123rd digit, after the first four, we have 119 digits in the cycle. 119 divided by 3 is 39 with a remainder of 2, so the second digit in "189" is 8.Therefore, the answer is 8.

💡Okay, so I need to figure out the percent decrease in the price of these trousers that John wanted to buy. Let me start by understanding what percent decrease means. It's basically how much the price has gone down, expressed as a percentage of the original price. Alright, the original price of the trousers was 100, and then it went on sale for 40. So, the first thing I should do is find out how much the price actually decreased. To do that, I'll subtract the sale price from the original price. So, 100 minus 40 equals 60. That means the price decreased by 60. Now, to find out what percentage this decrease is relative to the original price, I need to divide the decrease by the original price. So, 60 divided by 100 equals 0.6. Hmm, that's a decimal, but I need a percentage. To convert that decimal to a percentage, I multiply it by 100. 0.6 times 100 equals 60%. So, the price decreased by 60%. That seems pretty significant! Wait, let me double-check my calculations to make sure I didn't make any mistakes. Original price minus sale price is 60, and 60 divided by 100 is indeed 0.6. Multiplying by 100 gives me 60%. Yeah, that looks correct. I guess another way to think about it is, if the original price was 100 and it's now 40, then the price is 40% of the original price. So, the decrease would be 100% minus 40%, which is also 60%. That matches what I calculated earlier. Just to be thorough, I'll try plugging in some other numbers to see if this makes sense. If something was 200 and went down to 80, the decrease would be 120, and 120 divided by 200 is 0.6, which is also 60%. So, it seems like the method works consistently. I think I've got it. The percent decrease is 60%.

💡Okay, so I have this math problem here: "If one-fourth of (2^{30}) is equal to (2^x), what is (x)?"Alright, let's break this down. First, I need to understand what "one-fourth of (2^{30})" means. "One-fourth of something" is the same as multiplying that something by (frac{1}{4}). So, in mathematical terms, this would be (frac{1}{4} times 2^{30}).Now, I remember that fractions can be expressed as exponents. Specifically, (frac{1}{4}) is the same as (4^{-1}), right? But wait, (4) is (2^2), so (4^{-1}) would be ((2^2)^{-1}), which simplifies to (2^{-2}). So, (frac{1}{4}) is (2^{-2}).So now, the expression becomes (2^{-2} times 2^{30}). I recall that when you multiply exponents with the same base, you add the exponents. So, (2^{-2} times 2^{30} = 2^{-2 + 30}).Calculating the exponent: (-2 + 30 = 28). So, this simplifies to (2^{28}).The problem states that this is equal to (2^x). So, (2^{28} = 2^x). Since the bases are the same, the exponents must be equal. Therefore, (x = 28).Wait, let me double-check to make sure I didn't make a mistake. Starting from the beginning: one-fourth of (2^{30}) is (frac{1}{4} times 2^{30}). Converting (frac{1}{4}) to a power of 2 gives (2^{-2}). Multiplying (2^{-2}) by (2^{30}) adds the exponents: (-2 + 30 = 28). So, (2^{28} = 2^x) implies (x = 28). Yep, that seems correct.I think I've got it. The value of (x) is 28.

💡First, I need to determine which of the given points lies on the parabola defined by the equation ( y = frac{1}{2}x^2 - 2 ).I'll start by checking the points where ( x = 0 ), which are options A and D.For option A: ( (0, 2) )Substitute ( x = 0 ) into the equation:[y = frac{1}{2}(0)^2 - 2 = 0 - 2 = -2]This does not match the ( y )-value of 2, so option A is incorrect.For option D: ( (0, -4) )Substitute ( x = 0 ) into the equation:[y = frac{1}{2}(0)^2 - 2 = 0 - 2 = -2]This does not match the ( y )-value of -4, so option D is also incorrect.Next, I'll check the points where ( y = 0 ), which are options B and C.For option B: ( (2, 0) )Substitute ( x = 2 ) into the equation:[y = frac{1}{2}(2)^2 - 2 = frac{1}{2}(4) - 2 = 2 - 2 = 0]This matches the ( y )-value of 0, so option B is correct.For option C: ( (4, 0) )Substitute ( x = 4 ) into the equation:[y = frac{1}{2}(4)^2 - 2 = frac{1}{2}(16) - 2 = 8 - 2 = 6]This does not match the ( y )-value of 0, so option C is incorrect.Therefore, the parabola passes through the point ( (2, 0) ), which corresponds to option B.

💡First, I need to evaluate the expression (2^3 - 2 + 3).I start by calculating (2^3):[2^3 = 2 times 2 times 2 = 8.]Next, I substitute (8) back into the expression:[8 - 2 + 3.]Then, I perform the subtraction:[8 - 2 = 6.]Finally, I add (3) to (6):[6 + 3 = 9.]Therefore, the value of the expression is (9).