Test 3 Section 2 Part 5

TL;DR

This SAT Math practice covers problems on arithmetic mean, geometry, and remainders.

Transcript

Problem number 18. The average arithmetic mean of x and y is k. So the average, so x plus y/2 is k, right? That's the average of x and y is equal to k. Which of the following is the average of x, y and z? So they want to know what x plus y-- so the average of x, y, and z is, you add up all the numbers and you divide by 3. They want to know in terms... Read More

Key Insights

• #️⃣ The average of three numbers can be found by substituting the appropriate value for the average of two numbers.
• 🔺 In an equilateral triangle, all angles are 60 degrees, and splitting a 90-degree angle results in two 30-degree angles.
• 🔺 The side opposite a 60-degree angle in a 30-60-90 triangle is equal to the square root of 3 times the shorter leg.
• 😉 The remainder when dividing 15 by a positive integer k can be found by testing different values of k. In this case, k = 4, 6, and 12.

Questions & Answers

Q: What is the average of x, y, and z in terms of k and z?

The average is 2k + z/3. This can be obtained by substituting 2k for x + y in the formula for the average of three numbers.

Q: How do we know the triangle in the second problem is a 30-60-90 triangle?

Since the angle opposite the diameter of the circle is a right angle, and the other angles of the equilateral triangle are 60 degrees each, the remaining angle is split into two 30-degree angles.

Q: What is the radius of the circle in the second problem?

The radius is equal to half the length of the diameter, which is found to be the square root of 3/2.

Q: How many values of k make 15 divided by k have a remainder of 3?

The values 12, 6, and 4 are the three different values of k that satisfy this condition.

Summary & Key Takeaways

• The first problem requires finding the average of three numbers, given the average of two numbers.

• The second problem asks for the area of a circle inscribed in an equilateral triangle.

• The third problem involves finding the number of different values of k that give a remainder of 3 when 15 is divided by k.