## In a certain class of 70 students, 4/7 of the students are boys, and the ratio of students 10 years or older to students less than 10 years is 2:3. If 2/3 of the girls are less than 10 years old, how many boys are 10 years old or older?

OK, so you have 40 boys and 30 girls. That’s easy enough to calculate because you’re given a part:whole ratio (boys to total students) and you already know the total number of students is 70. Be careful about the second ratio, though, because it’s a part:part ratio! If the ratio of older to younger is (more…)

## The polygon ABCDEFGHI is a regular 9-sided polygon with consecutive vertices labeled in alphabetical order. What is the measure of angle ACE?

OK, so when you have a regular n-gon, you can figure out each angle in it using this formula: [(n-2)180]/n. In this case, 7*180/9 = 140, so we know each angle in the polygon is 140°. I couldn’t draw this quickly on the computer I’m on, so I found a good n-gon picture to mark (more…)

## In the xy-plane, the point (6,3) is the midpoint of the line segment with endpoints (x,5) and (9,y). What is the value of x+y ?

The midpoint formula tells you that the a segment with endpoints (a, b) and (c, d) will have a midpoint at ((a + c)/2,(b + d)/2) So we know that (x + 9)/2 = 6 and (5 + y)/2 = 3. We can solve those! (x + 9)/2 = 6 x + 9 = 12 x = 3 (5 + y)/2 = 3 5 + y = (more…)

## 2x-5y=8 4x+ky=17 For which of the following values of k will the system of equations above have no solution (A)-10 (B)-5 (C)0 (D)5 (E)10

When a system of linear equations has no solution, that means you have parallel lines, which means the lines have the same slope. So put both equations into slope-intercept form (y = mx + b) first: In order for those lines to be parallel, their slopes must be equal, which means 2/5 = -4/k. That means k must be (more…)

## The volume, V, of a right circular cylinder is given by the formula V=πr^2*h, where r is the radius of the base and h is the height of the cylinder. If r is divided by 2 and h is multiplied by 2, the new cylinder’s volume V will be: A) 1/4 the original volume B) 1/2 the original volume C) the original volume D) 2 times the original volume

Just do those things and simplify: Looks like the volume of the second cylinder is half the volume of the first! 🙂 The shortcut here is to recognize that the r value gets squared but the h value doesn’t, so you can say the volume is divided by 4 then multiplied by 2. from Tumblr https://ift.tt/2De8uzO

## If the circumference of the circle with center O is 12π and angle AOB is 60 degree , what is the length of minor arc AB?

60° is 1/6 of the circle (which has 360° of arc in total), so the length of the minor arc will be 1/6 of the circumference. 1/6 of 12π is 2π. from Tumblr https://ift.tt/2JsMj92

## What is the x-coordinate of the vertex of the graph of y = -6x^2 + 3x + 8

Huge shortcut here if you just know that for a parabola in standard ax^2 + bx + c form, the x-coordinate of the vertex will be at –b/(2a). In this case, that means it’s at –3/(2(–6)) = 3/12 = ¼. from Tumblr https://ift.tt/2CRW93a

## If f(x) = 2x + 3 and f(g(x)) = 8x – 1, which of the following equals g(x)? A) 4x-2 B) 4x-1 C) 8x-4 D) 8x-1 E) 16x+23

The way to think about this (for me, anyway) begins with understanding that 2[something] + 3 = 8x – 1, and our job is to figure out what that something is. Since this question gives us answer choices, all we really need to do is try each one as the something to see what works. Since (more…)

## (a^2x) ÷ (a^(x-y)) = a, then y equals : A) -x B) 1 C) x D) x-1 E) 1-x

When you have exponents with the same base and you divide them, you subtract the exponents. So from what we’re given, we know that 2x – (x – y) = 1. We can solve that for y. 2x – (x – y) = 1 2x – x + y = 1 x + y = 1 y = (more…)

## In a circle with center O, the length of radius OB is 5. If the length of chord CD is 8, and if line CD perpendicular to line OB, then sin angle OCD =

Start by drawing it! Note that OC = 5 and OD = 5 because both of those are also radii. Note also that because chord CD is perpendicular to OB, it’s bisected by OB. In other words, it’s split into 2 segments each measuring 4.   Things are really coming together! Because we know our Pythagorean (more…)

## If (x+7)(x-3) = 0, what are all possible values of x-3 ?

One of the two things being multiplied must be zero, so either x – 3 = 0 or x + 7 = 0. If x + 7 = 0, then x = –7. In that case, we can substitute into x – 3: x – 3 = –7 – 3 = –10 So x – 3 can equal 0 or –10. from Tumblr (more…)

## In the xy-plane, the coordinates of the vertices of triangle ABC are A(-8,0), B(0,4), and C(x,0). If AB = BC, what is the value of x?

Draw! You know point C must be on the x-axis, and that AB = BC. Since B is on the y-axis, it turns out all we need to do here is reflect point A across the y-axis. Point C must be at (8, 0), so x = 8. from Tumblr https://ift.tt/2yLFCer

## Which of the following statements is true for all real numbers c ? A) 8+c > 4+c B) 8+4c > 4-4c C) 4c > 8c D) 8c > 4c E) 8c^2 > 4c^2

The correct answer is A. Subtract c from both sides of the inequality and you’re left with 8 > 4. That’s always true! from Tumblr https://ift.tt/2Oa6SYI

## How many milliliters of water must be evaporated from 200 milliliters of a 30% saline solution to obtain a 40% saline solution?

In 200 ml of 30% saline, there’s (0.3)(200) = 60 ml of saline. 60 ml is 40% of what? 60 = (0.4)x 150 = x So the saline will be 40% when enough water evaporates that there’s 150 ml of solution. Therefore, 50 ml of the original 200 ml must evaporate. from Tumblr https://ift.tt/2CAqROd

## While flying from Los Angeles to New york, Pat fell asleep after travelling exactly half of the distance. When she woke up, tge distance remaining was half of the distance she travelled while asleep. For what part of the trip was pat asleep?

Let’s work backwards, and use hours instead of distance for ease (assume the plane travels at a constant speed for our purposes). Let’s say when she woke up, she had 1 hour left in her flight. That’s half of the time she was sleeping so she must’ve slept for 2 hours. She first fell asleep (more…)