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Algebra Tools: Place-Out Exam

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1.

What 2-digit number has the property that if you subtract 5 times its tens digit from it, the result is the 2-digit number with the digits swapped?

2.

Suppose that the symbol x @ yx \ @ \ y means:

1x+1y. \frac{1}{x} + \frac{1}{y}.

If 2 @ (3 @ x)=4,2 \ @ \ (3 \ @ \ x) = 4,

what is x?x?

3.

What is the sum of all odd 2-digit numbers?

4.

Starting from a point on the ground, a ball is thrown straight up to a height of 11 meter. It bounces up and down vertically, each time reaching a maximum height that is 56\frac{5}{6} its previous maximum height. How many total meters will the ball travel, if it keeps bouncing forever?

5.

Simplify

15+414 \sqrt{15 + 4 \sqrt{14}}

and rewrite it in simplest radical form as

ab+c,a \sqrt{b} + \sqrt{c},

where a,a, b,b, and cc are integers. What is the value of a+b+c?a + b + c?

6.

Elijah counts 3131 creepy-crawlers, all of which are spiders (which have 88 legs) or ants (which have 66) legs. The creepy-crawlers have 214214 legs altogether. How many spiders are there?

7.

Find the value of the positive number:

2+2+2+ \sqrt{2 + \sqrt{2 + \sqrt{2 + \cdots}}}

8.

A square's area plus its perimeter equals 221.221. What is the side length of the square?

9.

A straight line passes through the points

(10,13) \left( -10, \frac{1}{3} \right)

and

(2,173). \left( -2, \frac{17}{3} \right).

What is the yy-intercept of another line perpendicular to this first line which passes through the midpoint between

(10,13) \left( -10, \frac{1}{3} \right)

and

(2,173)? \left( -2, \frac{17}{3} \right)?

10.

A triangle has corners A(0,0),A(0, 0), B(12,0),B(12, 0), and C(0,6).C(0, 6). If the midpoint of ABAB is connected to CC and the midpoint of ACAC is connected to B,B, they intersect at a point with coordinates (x,y).(x, y). What is y?y?