AUTO ANSWER (AFTER SEVERAL DAYS):

Since the two acute angles in a right triangle are complementary, we know that the measure of angle APB must be 90 – 17 = 73 degrees. We also know that the measure of angle APQ is 60 degrees. Angle BPC is a straight angle, so the measure of angle QPC is 180 – 73 – 60 = 47 degrees.

AUTO ANSWER (AFTER SEVERAL DAYS):

 The letter m in the equation y = mx + b represents the slope of the line. We need to calculate the ratio of the change in the y-values to the change in the x-values. Using the points (6, 13) and (10, 31), we have a slope of m = (31 – 13)/(10 – 6) = 18/4 = 9/2.

AUTO ANSWER (AFTER SEVERAL DAYS):

The least common multiple (LCM) of 12 and 20 is 60. So let the number of 12-ounce cans of soda that Dewey buys be 5, and let the number of 20-ounce bottles of soda that Peppar buys be 3. They each get a total of 60 ounces, but Dewey spends 5 × 1.00 = $5.00, and Peppar spends 3 ×1.25 = $3.75. Peppar spends 5.00 – 3.75 = $1.25 less than Dewey, which is 1.25 ÷ 5.00 × 100 = 25, so P = 25.

AUTO ANSWER (AFTER SEVERAL DAYS):

The area of the entire dartboard is the area of the circle of radius 10 inches, or π × 10\(^2\) = 100π in\(^2\). The area of the yellow region of the dart board is the difference between the areas of the circle of radius 5 inches and the circle of radius 1 inch, or π × 5\(^2\) – π × 1\(^2\) = 25π – π = 24π in\(^2\). Thus, 24π/(100π) = 6/25 of the dartboard’s total area is colored yellow, and that fraction is the desired probability.  

We have: 4 apples + 6 bananas = 1.56

                9 apples + 7 bananas = 2.60

=> 4 apples + 6 bananas + 9 apples + 7 bananas = 1.56 + 2.60

13 apples + 13 bananas = 4.16

=> 1 apple + 1 banana = \(4.16\div13=0.32\)

So the answer is: 0.32

P/s: Sorry I don't know how to do the euro sign.

The time to travel upstream for 10 miles was 10/(20 − 5) = 10/15 = 2/3 hour, or 40 minutes. So, of the entire 64-minute round-trip, the fraction that was spent traveling upstream was 40/64 = 5/8.

ANSWER:

Let c represent the speed of the river’s current. Then Ansel’s speed was 20 + c downstream and 20 − c upstream. The time to travel the 10 miles downstream and then back upstream was 10/(20 + c) + 10/(20 − c). Since we know that the entire round-trip took 64 minutes, or 16/15 hours, we have 10/(20 + c) + 10/(20 − c) = 16/15. Solving for c, we find that (200 + 10c + 200 − 10c)/(400 − c\(^2\)) = 16/15 → 400/(400 − c\(^2\)) = 16/15 → 15(400) = 16(400 − c\(^2\)) → 6000 = 6400 − 16c\(^2\) → 16c\(^2\) = 400 → c\(^2\) = 25 → c = 5. Thus, the speed of the river’s current is 5 mi/h.

Sorry, N doesn't include 9 (line 3)

\(N⋮99\)

=> \(N\in\left\{99;198;297;...;1089;1188;...\right\}\)

Because N doesn't include N 

=> The smallest positive integers N = 1188.

ANSWER:

The difference between the first and second terms is 2x + 11 − x = x + 11. The difference between the second and third terms is 4x − 3 − (2x +11) = 4x − 3 − 2x −11 = 2x − 14. Since the difference between consecutive terms is constant, we set these two differences equal to each other and get x + 11 = 2x − 14 → x = 25. Substituting, we see that the difference between consecutive terms is x + 11 = 25 + 11 = 36.

ANSWER:

Based on the information provided, we can find f\(^5\)(x) in a number of ways. For example, f\(^5\)(x) = f\(^2\)(f\(^3\)(x)) = f\(^3\)(f\(^2\)(x)) = f\(^4\)(f(x)) = f(f\(^4\)(x)) = ax + b. We can easily derive either f\(^3\)(x) or f\(^4\)(x) using the information provided in the problem. We have f\(^3\)(x) = f\(^2\)(f(x)) = 4(2x + 3) + 9 = 8x +12 + 9 = 8x + 21. So, f\(^5\)(x) = f\(^2\)(f\(^3\)(x)) = 4(8x + 21) + 9 = 32x + 84 + 9 = 32x + 93. We see that a = 32, b = 93 and a + b = 32 + 93 = 125.

ANSWER:

Let x, rx, r\(^2\)x, and r\(^3\) xrepresent the degree measures of the angles of this quadrilateral. If r = 2, our angle measures are x, 2x, 4x and 8x. Then x + 2x + 4x + 8x = 360 → 15x = 360 → x = 24. The largest angle has degree measure 8 × 24 = 192 degrees. If r = 3, our angle measures are x, 3x, 9x and 27x. Then x + 3x + 9x + 27x = 360 → 40x = 360 → x = 9. The largest angle has degree measure 27 × 9 = 243 degrees. Letting r = 4, or any higher value, yields non-integer angle measures. So, the largest possible degree measure of an angle in this quadrilateral is 243 degrees.

To determine the average speed for the entire round-trip, we need the total distance and the total time. If Jack and Jill travel d miles uphill, then they travel another d miles downhill, for a total distance of 2d miles. Since time = distance/speed, it follows that the time to travel uphill is d/2 hours and the time to travel downhill is d/4 hours, for a total time of d/2 + d/4 = (2d + d)/4 = 3d/4 hours. Now we can use these values in the formula speed = distance/time to get 2d/(3d/4) = 2d × 4/(3d) = 8/3 =  \(2\dfrac{2}{3}\)mi/h.

ANSWER:

Since distance = speed × time, we know that when they meet in t hours, Jack will have traveled 4t miles and Jill will have traveled 2t miles. We also know that the total distance traveled is 1.5 miles. Therefore, 4t + 2t = 1.5 → 6t = 1.5 → t = 1/4 hour, or 15 minutes. They will meet at 2:35 p.m.

ANSWER: 

We need to determine how far Jill has traveled up the hill when she meets Jack. Jill traveled at a speed of 2 mi/h for 1/4 hour. That’s a total distance of 2 × 1/4 = 1/2 mile = 1/2 × 5280 × 1/3 = 1760/2 = 880 yards.

ANSWER:
We know that Alysha drives to the market at a speed of 2/3 mile per minute and it takes her 3 minutes. That means the distance from her home to the market is 2/3 × 3 = 2 miles.

We have: 

- row 1 = \(1^3=1\) 

- row 2 = \(2^3=8\) 

- row 3 = \(3^3=27\) 

...........

- row 20 = \(20^3=8000\) 

So the answer is 8000.

The number is divisible by 11.

Suppose it is a two digit number (The number is aa)

=> aa is not a gallant number because \(aa⋮̸a+a\)

If the number is 110 => Unsatisfy because 110 \(⋮̸\)0

If the number is 121 => Unsatisfy because 121 \(⋮̸\)4

If the number is 132 => Satisfy

So the smallest number satisfy the question is 132