Practice (TheColoringMethod)

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What fraction of the first 100 triangular numbers are evenly divisible by 7? Express your answer as a common fraction.

The analog clock shown has a minute hand with an arrow tip that is exactly twice as far from the clock\u2019s center as the hour hand\u2019s arrow tip. If point A is at the tip of the minute hand, and point B is at the tip of the hour hand, what is the ratio of the distance that point B travels in 3 hours to the distance that point A travels in 9 hours? Express your answer as a common fraction.


What percent of the interval with endpoints \u22125 and 5 consists of real numbers $x$ satisfying the inequality $x + 1 > \frac{8}{x -1}$?

In the figure shown here, the distance between any two horizontally or vertically adjacent dots is one unit. What is the area of the shaded polygon? Express your answer as a decimal to the nearest tenth.


Barbara completely fills her mug with a mixture that is 15 mL of hot chocolate and 35 mL of cream. What percent of the mixture is hot chocolate?

A certain lottery has 20 million possible combinations in their weekly drawing. If you wanted to purchase 20 million tickets, how many tickets would you need to purchase each second, on average, to buy them all in one week? Express your answer to the nearest whole number.

What is the mean of all possible positive three-digit integers in which no digit is repeated and all digits are prime? Express your answer as a decimal to the nearest hundredth.

Ray\u2019s age is half his sister's age, and his age is the square root of one-third their grandfather's age. In 5 years, Ray will be two-thirds as old as his sister will be then. What is the ratio of Ray's sister's age to their grandfather's age right now? Express your answer as a common fraction.

Alex added the page numbers of a book together and got a total of 888. Unfortunately, he didn't notice that one of the sheets of the book was missing with an odd page number on the front and an even page number on the back. What was the page number on the final page in the book?

A circular spinner has seven sections of equal size, each of which is colored either red or blue. Two colorings are considered the same if one can be rotated to yield the other. In how many ways can the spinner be colored?


A square is inscribed in a circle of radius 5 units. In each of the four regions bounded by a side of the square and the smaller circular arc joining the endpoints of that side, a square is drawn so that one side lies on the side of the larger square and the two opposite vertices lie on the circle, as shown. What is the total area of the five squares? Express your answer to the nearest whole number.


At 840,000 $mile^2$, Greenland is three times the size of Texas. By comparison, Iceland is only 39,800 $mile^2$. What percent of Texas would be covered by Iceland? Express your answer as a decimal to the nearest tenth.

A square and a regular hexagon are coplanar and share a common side as shown. What is the sum of the degree measures of angles 1 and 2?


A field trip costs \$9000 for the bus, plus \$125 per student. Each student pays $375. How many students must go on the trip so that the total amount paid is equal to the total cost of the trip?

$\triangle{ABC}$ has vertices at A(-3, 4), B(5, 0) and C(1, -4). What is the $x$-coordinate of the point where the median from C intersects $\overline{AB}$?

The sum of three primes is 125. The difference between the largest and the smallest is 50. What is the largest possible median of these three prime numbers?

What is the probability that a randomly selected integer from $1$ to $81$, inclusive, is equal to the product of two one-digit numbers?

Shaina has one stick of length $a$ cm and another of length $b$ cm, where $a \ne b$. She needs a third stick with length strictly between 8 cm and 26 cm to make the third side of a triangle. What is the product $ab$?

A 3-inch by 8-inch sheet of paper and a 2-inch by 12-inch sheet of paper have the same area. Using just one cut (not necessarily straight), the 3-inch by 8-inch sheet can be divided into two pieces that can be rearranged to completely cover the 2-inch by 12-inch sheet. What is the length of the cut?

Suppose $d = 3 + 33 + 303 + 3003 + 30,003 + \cdots$, where each addend after the second has one more "interior" 0 than the previous addend. If the last addend has thirty digits, what is the sum of the digits of $d$?

A bullet train traveling 210 km/h is catching up to a freight train traveling 90 km/h on a parallel track. From the time the front of the bullet train catches up to the back of the freight train to the time the back of the bullet train pulls even with the front of the freight train 24 seconds elapse. If freight trains are three times as long as bullet trains, how many seconds would it take two bullet trains, each traveling at 210 km/h, to pass by each other completely when moving in opposite directions? Express your answer as a decimal to the nearest hundredth.

A bucket with a hole is filled with 10 gallons of water, but loses 2 gallons every 10 minutes. How many minutes does it take for the bucket to be empty?

The sum of the digits of a two-digit positive integer is 9. The difference between this integer and a two-digit integer containing the digits reversed is 27. Find the product of the two digits.

How many diagonals does a convex octagon have?

Jean is twice as likely to make a free throw as she is to miss it. What is the probability that she will miss $3$ times in a row?