💡First, I need to understand the problem statement. It says that a number twelve times as large as ( x ) is increased by five, and then one third of the result is equal to what value in terms of ( x ).To break it down, I'll start by representing the number twelve times as large as ( x ) as ( 12x ). Next, I'll increase this number by five, which gives me the expression ( 12x + 5 ).Now, the problem asks for one third of this result. So, I'll take the expression ( 12x + 5 ) and multiply it by ( frac{1}{3} ). This gives me ( frac{1}{3}(12x + 5) ).To simplify this, I'll distribute the ( frac{1}{3} ) across both terms inside the parentheses. Multiplying ( 12x ) by ( frac{1}{3} ) gives ( 4x ), and multiplying ( 5 ) by ( frac{1}{3} ) gives ( frac{5}{3} ).Putting it all together, the simplified expression is ( 4x + frac{5}{3} ).Finally, I'll compare this result with the given options to find the correct answer.

💡First, I need to determine the total cost of the lunch before the 15% gratuity was added. Let’s denote the pre-gratuity total as ( x ). The gratuity would then be ( 0.15x ). The sum of the pre-gratuity total and the gratuity equals the total cost of 207.00.So, the equation is:[ x + 0.15x = 207 ]Combining like terms gives:[ 1.15x = 207 ]To find ( x ), I divide both sides by 1.15:[ x = frac{207}{1.15} ]Calculating this:[ x = 180 ]The total cost before gratuity is 180.00. To find the average price per person excluding gratuity, I divide this amount by the number of people:[ text{Average price per person} = frac{180}{15} = 12 ]Therefore, the average price per person, excluding the gratuity, is 12.00.

💡First, identify the principal amount, which is Rs. 2000.Calculate the interest B pays to A annually:15% of 2000 = 300 Rs.Calculate the interest B earns from C annually:17% of 2000 = 340 Rs.Determine B's annual gain:340 Rs. (from C) - 300 Rs. (to A) = 40 Rs.Find the number of years for the total gain of Rs. 160:160 Rs. ÷ 40 Rs. per year = 4 years.

💡First, I need to evaluate the expression ([-9 Delta (-3)] times [4 Delta (-3)]) using the defined operation (Delta).1. **Evaluate (-9 Delta (-3)):** - Compare (-9) and (-3). Since (-9 leq -3), I use the rule (x Delta y = sqrt{|x|}). - Calculate (sqrt{|-9|} = sqrt{9} = 3).2. **Evaluate (4 Delta (-3)):** - Compare (4) and (-3). Since (4 > -3), I use the rule (x Delta y = y). - Therefore, (4 Delta (-3) = -3).3. **Multiply the results:** - Multiply (3) by (-3) to get (3 times (-3) = -9).Thus, the final value of the expression is (-9).

💡First, calculate the total number of possible three-digit codes without any restrictions. Each digit can be any number from 0 to 9, so there are (10^3 = 1000) possible codes.Next, determine the number of codes that Tanya cannot choose. These include:1. Codes that differ from 314 in only one digit position. For each of the three digit positions, there are 9 possible replacements (excluding the original digit). This results in (3 times 9 = 27) codes.2. Codes that are permutations of 314 by swapping two digits. There are 3 such permutations: 134, 413, and 341.3. The code 314 itself.Adding these together, Tanya cannot choose (27 + 3 + 1 = 31) codes.Finally, subtract the number of restricted codes from the total number of possible codes to find the number of available codes for Tanya:[1000 - 31 = 969]

💡Okay, so I have this polynomial: x³ + b₂x² + b₁x - 30 = 0. I need to find all the possible integer roots of this polynomial. Hmm, I remember something about the Integer Root Theorem, or maybe it's called the Rational Root Theorem? Let me think. I think it's the Rational Root Theorem, but since we're only interested in integer roots, that should simplify things.Alright, the Rational Root Theorem says that any possible rational root, expressed in lowest terms p/q, p is a factor of the constant term, and q is a factor of the leading coefficient. In this case, the polynomial is a cubic, and the leading term is x³, so the leading coefficient is 1. That means q must be ±1, right? So, any rational root is just an integer that divides the constant term, which is -30.So, the possible integer roots are all the divisors of -30. Let me list them out. The divisors of 30 are 1, 2, 3, 5, 6, 10, 15, 30. But since it's -30, the negative counterparts are also divisors. So, the possible integer roots should be ±1, ±2, ±3, ±5, ±6, ±10, ±15, ±30.Wait, let me make sure I'm not missing anything. The polynomial is x³ + b₂x² + b₁x - 30. The constant term is indeed -30, and the leading coefficient is 1. So, according to the theorem, all possible integer roots are the divisors of -30, which are the numbers I listed.But just to double-check, maybe I should plug in some of these values into the polynomial to see if they could be roots. Let's take x=1: 1 + b₂ + b₁ - 30. That simplifies to (1 + b₂ + b₁) - 30. Without knowing b₂ and b₁, I can't compute the exact value, but if 1 is a root, then the polynomial evaluated at 1 should be zero. So, 1 + b₂ + b₁ - 30 = 0, which means b₂ + b₁ = 29. That's possible, but I don't know if it's the case here.Similarly, if x=2: 8 + 4b₂ + 2b₁ - 30 = 0. Simplifying, 4b₂ + 2b₁ = 22, or 2b₂ + b₁ = 11. Again, possible, but without knowing the coefficients, I can't confirm.Wait, maybe I don't need to plug them in. The question is just asking for all possible integer roots, not necessarily which ones are actual roots. So, I think I was right the first time. The possible integer roots are all the divisors of -30, which are ±1, ±2, ±3, ±5, ±6, ±10, ±15, ±30.Just to make sure I didn't miss any divisors, let me list the factors of 30 again. 30 can be factored into 1×30, 2×15, 3×10, 5×6. So, the positive divisors are 1, 2, 3, 5, 6, 10, 15, 30. Including the negatives, that's ±1, ±2, ±3, ±5, ±6, ±10, ±15, ±30. Yep, that's all of them.I think I'm confident now that these are all the possible integer roots. The coefficients b₂ and b₁ are integers, so the polynomial has integer coefficients, which is a condition for the Integer Root Theorem to apply. So, yeah, the possible integer roots are all the divisors of -30, which are the numbers I listed.