Question

clt preparation mathematics / quantitative reasoning practice problems
please note this document has absolutely no official relationship with the classic learning test, these are merely mathematics problems i wrote in a similar style for practice.

1. sarah is organizing her home library. in one corner of the library, she has four small shelves, each of which can fit five books. she wants to put at least one novel, and one non - fiction book, on each shelf, and she has an abundance of each type of book. what is the maximum number of novels that could be placed on the four shelves?

a) 4
b) 12
c) 16
d) 20

1. how many integers between 5 and 55, inclusive, are divisible by 3?

a) 11
b) 12
c) 16
d) 17

1. the yard of a home is shaped like a square, 10 m on a side, with a semicircle affixed on the north side of it, as generally depicted below. what is the area of the yard?

a) 100 m²
b) 125 m²
c) 100 + 12.5π m²
d) 100 + 50π m²

Explanation:

Problem 1

Step1: Determine books per shelf after minimum non - fiction

Each shelf can hold 5 books, and we need at least 1 non - fiction book per shelf. So the number of novels per shelf at maximum is \(5 - 1=4\).

Step2: Calculate total novels for 4 shelves

There are 4 shelves. So the total number of novels is \(4\times4 = 16\).

Step1: Find first and last multiple

The first integer between 5 and 55 (inclusive) divisible by 3 is 6, and the last one is 54.

Step2: Use arithmetic sequence formula

The formula for the \(n\)th term of an arithmetic sequence is \(a_n=a_1+(n - 1)d\), where \(a_n = 54\), \(a_1=6\), \(d = 3\).
\(54=6+(n - 1)\times3\)
\(54-6=(n - 1)\times3\)
\(48=(n - 1)\times3\)
\(n - 1=\frac{48}{3}=16\)
\(n=17\)

Step1: Calculate area of square

The side of the square is \(10\) m. The area of a square is \(A_{square}=s^2\), so \(A_{square}=10^2 = 100\space m^2\).

Step2: Calculate area of semicircle

The diameter of the semicircle is equal to the side of the square, so \(d = 10\) m, radius \(r=\frac{10}{2}=5\) m. The area of a full circle is \(A_{circle}=\pi r^2\), so the area of a semicircle is \(A_{semicircle}=\frac{1}{2}\pi r^2=\frac{1}{2}\pi\times5^2=\frac{25\pi}{2}=12.5\pi\space m^2\).

Step3: Calculate total area of yard

The total area of the yard is the sum of the area of the square and the area of the semicircle, so \(A = 100+12.5\pi\space m^2\).

Answer:

C) 16

Problem 2