💡First, I need to determine the number of red colored pencils. Since red pencils make up 5/9 of the total 36 pencils, I can calculate this by multiplying 36 by 5/9.Next, I'll find the number of blue colored pencils. Blue pencils account for 5/12 of the total, so I'll multiply 36 by 5/12.After obtaining both numbers, I'll subtract the number of blue pencils from the number of red pencils to find out how many more red pencils there are compared to blue pencils.

💡First, I need to determine the total number of vehicles that traveled on the highway last year, given that 2880 vehicles were involved in accidents.The problem states that for every 100 million vehicles, 96 were involved in accidents. This means that the accident rate is 96 accidents per 100 million vehicles.To find out how many sets of 100 million vehicles correspond to 2880 accidents, I can set up a proportion:96 accidents / 100,000,000 vehicles = 2880 accidents / V vehiclesSolving for V, I get:V = (2880 accidents * 100,000,000 vehicles) / 96 accidentsV = 3,000,000,000 vehiclesTherefore, 3 billion vehicles traveled on the highway last year.

💡Okay, so I have this problem where I need to find the equation of a line that passes through the origin and is tangent to a given circle. The circle's equation is x² + y² - 6x + 5 = 0. Hmm, let me try to figure this out step by step.First, I remember that the general equation of a circle is (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius. But the given equation isn't in that standard form. Maybe I should rewrite it to find the center and radius.So, starting with x² + y² - 6x + 5 = 0. I think I need to complete the square for the x terms. Let me rearrange the equation:x² - 6x + y² + 5 = 0.Now, to complete the square for x² - 6x, I take half of -6, which is -3, square it to get 9, and add and subtract that inside the equation:(x² - 6x + 9) - 9 + y² + 5 = 0.Simplifying this, I get:(x - 3)² + y² - 4 = 0.Adding 4 to both sides:(x - 3)² + y² = 4.Okay, so the circle has center at (3, 0) and radius 2. Got that.Now, the line passes through the origin, so its equation should be y = mx, where m is the slope. Since it's tangent to the circle, the distance from the center of the circle to the line should be equal to the radius.I remember the formula for the distance from a point (h, k) to the line ax + by + c = 0 is |ah + bk + c| / sqrt(a² + b²). In this case, the line is y = mx, which can be rewritten as mx - y = 0. So, a = m, b = -1, c = 0.The center of the circle is (3, 0), so plugging into the distance formula:|m*3 + (-1)*0 + 0| / sqrt(m² + (-1)²) = |3m| / sqrt(m² + 1).This distance should equal the radius, which is 2. So,|3m| / sqrt(m² + 1) = 2.Since distance is always positive, I can drop the absolute value:3|m| / sqrt(m² + 1) = 2.But since m can be positive or negative, I can square both sides to eliminate the square root:(9m²) / (m² + 1) = 4.Multiplying both sides by (m² + 1):9m² = 4(m² + 1).Expanding the right side:9m² = 4m² + 4.Subtracting 4m² from both sides:5m² = 4.Dividing both sides by 5:m² = 4/5.Taking square roots:m = ±2/√5.Hmm, I can rationalize the denominator:m = ±(2√5)/5.So, the equations of the tangent lines are y = (2√5)/5 x and y = -(2√5)/5 x.Let me double-check if this makes sense. The slope is positive and negative, which means there are two tangent lines from the origin to the circle, one on each side. The radius is 2, and the center is at (3,0). The distance from (3,0) to the line y = mx should indeed be 2.Plugging m = 2√5/5 into the distance formula:|3*(2√5/5)| / sqrt((2√5/5)² + 1) = (6√5/5) / sqrt((4*5)/25 + 1) = (6√5/5) / sqrt(20/25 + 25/25) = (6√5/5) / sqrt(45/25) = (6√5/5) / (3√5/5) = 2. Perfect, that checks out.Similarly, for m = -2√5/5, the distance would also be 2 because of the absolute value. So, both lines are valid.Therefore, the equations of the tangent lines are y = ±(2√5)/5 x.**Final Answer**The equation of the line ( l ) is boxed{y = pm dfrac{2sqrt{5}}{5}x}.

💡Okay, so I have this problem about a 1/5-array, and I need to find the sum of all its terms. The array is structured in a specific way, with each row starting with a value that's 1/10 of the first value of the previous row. Then, each subsequent term in the row is 1/5 of the previous term in that row. The goal is to find the sum of all these terms and express it as a fraction m/n, where m and n are coprime, and then compute m + n modulo 100.Alright, let me try to visualize this array. The first row starts with 1, then each term is multiplied by 1/5, so it goes 1, 1/5, 1/25, 1/125, and so on. The second row starts with 1/10, then each term is multiplied by 1/5, so it's 1/10, 1/50, 1/250, etc. The third row starts with 1/100, then 1/500, 1/2500, and so on. Each new row starts with 1/10 of the first term of the row above it.So, if I think about the general term in the rth row and cth column, it should be the first term of the rth row multiplied by (1/5) raised to the power of (c - 1). The first term of the rth row is (1/10) raised to the power of (r - 1), right? Because each row starts with 1/10 of the previous row's first term.Let me write that down:The value at row r and column c is:[left(frac{1}{10}right)^{r - 1} times left(frac{1}{5}right)^{c - 1}]Wait, actually, looking back at the problem statement, it says the first entry of each row is 1/10 times the first entry of the previous row. So, the first term of the first row is 1. The first term of the second row is 1/10, the first term of the third row is 1/100, and so on. So, the first term of the rth row is (1/10)^{r - 1}.Then, each subsequent term in the row is multiplied by 1/5. So, the cth term in the rth row is (1/10)^{r - 1} multiplied by (1/5)^{c - 1}.Therefore, the general term is:[a_{r,c} = left(frac{1}{10}right)^{r - 1} times left(frac{1}{5}right)^{c - 1}]But in the problem statement, the first row starts with 1, so when r = 1, the first term is 1, which is (1/10)^{0} = 1. That makes sense. Similarly, for r = 2, it's (1/10)^{1} = 1/10, which matches the given array.Now, to find the sum of all terms in this array, I need to sum over all rows and all columns. That is, I need to compute:[sum_{r=1}^{infty} sum_{c=1}^{infty} a_{r,c}]Substituting the expression for a_{r,c}, we get:[sum_{r=1}^{infty} sum_{c=1}^{infty} left(frac{1}{10}right)^{r - 1} times left(frac{1}{5}right)^{c - 1}]Hmm, this looks like a double sum, which can be separated into the product of two geometric series. That is, since the terms are multiplicative over r and c, we can write this as:[left( sum_{r=1}^{infty} left(frac{1}{10}right)^{r - 1} right) times left( sum_{c=1}^{infty} left(frac{1}{5}right)^{c - 1} right)]Yes, that seems right. So, each of these sums is a geometric series.Let me recall that the sum of a geometric series starting from n = 0 is:[sum_{n=0}^{infty} ar^n = frac{a}{1 - r}]provided that |r| < 1.In our case, for the row sum, the first term when r = 1 is (1/10)^{0} = 1, so the series is:[sum_{r=1}^{infty} left(frac{1}{10}right)^{r - 1} = sum_{n=0}^{infty} left(frac{1}{10}right)^n = frac{1}{1 - 1/10} = frac{1}{9/10} = frac{10}{9}]Similarly, for the column sum, the first term when c = 1 is (1/5)^{0} = 1, so the series is:[sum_{c=1}^{infty} left(frac{1}{5}right)^{c - 1} = sum_{n=0}^{infty} left(frac{1}{5}right)^n = frac{1}{1 - 1/5} = frac{1}{4/5} = frac{5}{4}]Therefore, the total sum is:[frac{10}{9} times frac{5}{4} = frac{50}{36}]Simplify that fraction. Both numerator and denominator are divisible by 2:[frac{50 ÷ 2}{36 ÷ 2} = frac{25}{18}]So, the sum is 25/18. Now, m is 25 and n is 18. They are coprime because 25 is 5 squared, and 18 is 2 times 3 squared, so they share no common factors besides 1.Therefore, m + n is 25 + 18 = 43. The problem asks for m + n modulo 100. Since 43 is less than 100, 43 modulo 100 is just 43.Wait, let me double-check my steps to make sure I didn't make a mistake.1. Identified the general term correctly: (1/10)^{r - 1} * (1/5)^{c - 1}.2. Set up the double sum as the product of two geometric series.3. Calculated each geometric series correctly: 10/9 and 5/4.4. Multiplied them together to get 50/36, simplified to 25/18.5. Added m and n: 25 + 18 = 43.Everything seems to check out. I don't think I made any errors here. So, the final answer is 43 modulo 100, which is just 43.**Final Answer**boxed{43}

💡Okay, so I have this problem here: "Two dice are tossed. What is the probability that the sum is greater than four?" Hmm, let's see. I'm just starting to learn probability, so I need to break this down step by step.First, I know that when you toss two dice, each die has six faces, numbered from 1 to 6. So, the total number of possible outcomes when tossing two dice is 6 times 6, which is 36. That makes sense because for each of the six outcomes on the first die, there are six possible outcomes on the second die.Now, the problem is asking for the probability that the sum of the two dice is greater than four. So, I need to figure out how many of these 36 possible outcomes result in a sum greater than four.Maybe it's easier to first find the number of outcomes where the sum is four or less, and then subtract that from the total number of outcomes to find the number of outcomes where the sum is greater than four. That way, I can then calculate the probability by dividing by the total number of outcomes.Let's try that approach. So, what are the possible sums when two dice are tossed? The smallest possible sum is 2 (1+1), and the largest is 12 (6+6). But we're interested in sums that are four or less. So, let's list out all the possible outcomes that result in sums of 2, 3, or 4.For a sum of 2:- The only possible outcome is (1,1). So, that's 1 outcome.For a sum of 3:- The possible outcomes are (1,2) and (2,1). That's 2 outcomes.For a sum of 4:- The possible outcomes are (1,3), (2,2), and (3,1). That's 3 outcomes.Adding these up: 1 (for sum of 2) + 2 (for sum of 3) + 3 (for sum of 4) = 6 outcomes where the sum is four or less.So, there are 6 outcomes where the sum is four or less out of a total of 36 possible outcomes. Therefore, the number of outcomes where the sum is greater than four is 36 - 6 = 30.Now, to find the probability, I divide the number of favorable outcomes by the total number of possible outcomes. So, the probability of getting a sum greater than four is 30/36.I can simplify this fraction by dividing both the numerator and the denominator by 6. 30 divided by 6 is 5, and 36 divided by 6 is 6. So, the simplified probability is 5/6.Wait, let me double-check that. If there are 6 outcomes where the sum is four or less, then 36 - 6 is indeed 30, and 30 divided by 36 simplifies to 5/6. That seems correct.Alternatively, I could have listed all the outcomes where the sum is greater than four and counted them, but that would take longer since there are more such outcomes. So, calculating the complement (i.e., the probability of the sum being four or less) and subtracting from 1 seems more efficient.Just to be thorough, let's briefly consider listing the outcomes where the sum is greater than four. Starting from sum of 5 up to 12:For sum of 5:- (1,4), (2,3), (3,2), (4,1) → 4 outcomesFor sum of 6:- (1,5), (2,4), (3,3), (4,2), (5,1) → 5 outcomesFor sum of 7:- (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) → 6 outcomesFor sum of 8:- (2,6), (3,5), (4,4), (5,3), (6,2) → 5 outcomesFor sum of 9:- (3,6), (4,5), (5,4), (6,3) → 4 outcomesFor sum of 10:- (4,6), (5,5), (6,4) → 3 outcomesFor sum of 11:- (5,6), (6,5) → 2 outcomesFor sum of 12:- (6,6) → 1 outcomeAdding these up: 4 + 5 + 6 + 5 + 4 + 3 + 2 + 1 = 30 outcomes. Yep, that matches the earlier calculation. So, 30 outcomes where the sum is greater than four.Therefore, the probability is 30/36, which simplifies to 5/6. So, the probability of getting a sum greater than four when tossing two dice is 5/6.I think that's it. I've considered both approaches—calculating the complement and listing all favorable outcomes—and both lead to the same answer. That gives me confidence that 5/6 is the correct probability.

💡Okay, so I have this geometry problem here about angles formed by two lines intersected by a third line. The question is asking which of the given statements is incorrect. There are four options: A, B, C, and D. Let me try to understand each statement one by one.Starting with **Statement A: Vertical angles are equal.** I remember that vertical angles are the angles opposite each other when two lines intersect. They are always equal because they are congruent. So, this statement seems correct to me.Moving on to **Statement B: Adjacent angles are complementary.** Hmm, adjacent angles are angles that share a common side and a common vertex. The term "complementary" means that their measures add up to 90 degrees. But wait, isn't that only true if they form a right angle together? If two adjacent angles form a straight line, they would actually be supplementary, meaning their measures add up to 180 degrees. So, this statement might not always be true. It depends on the specific situation. Maybe this is the incorrect statement?Next is **Statement C: Corresponding angles are equal.** Corresponding angles are angles that are in the same relative position at each intersection where a transversal crosses two lines. I recall that corresponding angles are equal only if the two lines are parallel. The problem doesn't specify that the two lines are parallel, so this statement might not always be true. This could also be the incorrect statement.Lastly, **Statement D: If alternate interior angles are equal, then corresponding angles are also equal.** Alternate interior angles are angles on opposite sides of the transversal and inside the two lines. If alternate interior angles are equal, it implies that the two lines are parallel. And if the lines are parallel, then corresponding angles are also equal. So, this statement seems correct.Now, I need to figure out which one is the incorrect statement. Both statements B and C seem to have conditions where they might not hold true. Statement B talks about adjacent angles being complementary, which is only true in specific cases, not universally. Statement C talks about corresponding angles being equal, which is only true if the lines are parallel.But the question is about angles formed by two lines intersected by a third line, without specifying that the two lines are parallel. So, in the general case, corresponding angles are not necessarily equal. Therefore, Statement C is incorrect.Wait, but Statement B also seems incorrect because adjacent angles aren't always complementary. They could be supplementary if they form a straight line. So, which one is the answer? The question is asking for the incorrect statement, and both B and C have issues. But in the context of two lines intersected by a third line, I think the focus is more on the properties related to parallel lines, making Statement C the incorrect one.