Question

1. one side of square abcd has a length of 12 meters. a certain rectangle whose area is equal to the area of abcd has a width of 8 meters. what is the length, in meters, of the rectangle? a. 12 b. 16 c. 18 d. 20 e. 24 2. four identical glasses are shown below. one glass is empty, and the other 3 glasses are \\( \frac { 1 } { 4 } \\) full. \\( \frac { 1 } { 2 } \\) full. and \\( \frac { 4 } { 5 } \\) full of water, respectively. if the water were redistributed equally among the 4 glasses, what fractional part of each glass would be filled? e. \\( \frac { 2 } { 11 } \\) g. \\( \frac { 8 } { 11 } \\) h. \\( \frac { 3 } { 32 } \\) j. \\( \frac { 31 } { 60 } \\) k. \\( \frac { 31 } { 80 } \\) 3. the average of a list of 4 numbers is 92.0. a new list of 4 numbers has the same first 3 numbers as the original list, but the fourth number in the original list is 40, and the fourth number in the new list is 48. what is the average of this new list of numbers? f. 81.0 g. 92.0 h. 94.0 j. 94.4 k. 96.6

Explanation:

Question 1

Step1: Calculate the area of the square

The area formula for a square is \(A = s^2\) (where \(s\) is the side - length). Given \(s = 12\) meters, then \(A=12^2=144\) square meters.

Step2: Use the area formula for the rectangle

The area formula for a rectangle is \(A = l\times w\) (where \(l\) is the length and \(w\) is the width). We know \(A = 144\) square meters and \(w = 8\) meters. Rearranging the formula for \(l\), we get \(l=\frac{A}{w}\). Substituting the values, \(l=\frac{144}{8}=18\) meters.

Question 2

Step1: Calculate the total amount of water

The amounts of water in the non - empty glasses are \(\frac{1}{4}\), \(\frac{1}{2}\), and \(\frac{4}{5}\). The total amount of water is \(T=\frac{1}{4}+\frac{1}{2}+\frac{4}{5}\). First, find a common denominator (20). Then \(T=\frac{1\times5}{4\times5}+\frac{1\times10}{2\times10}+\frac{4\times4}{5\times4}=\frac{5 + 10+16}{20}=\frac{31}{20}\).

Step2: Redistribute the water among 4 glasses

Since we are redistributing the water among \(n = 4\) glasses, the fraction of each glass filled is \(F=\frac{T}{4}\). Substituting \(T=\frac{31}{20}\), we get \(F=\frac{31}{20}\div4=\frac{31}{20}\times\frac{1}{4}=\frac{31}{60}\).

Question 3

Step1: Find the sum of the original 4 numbers

The formula for the average \(\bar{x}=\frac{\sum_{i = 1}^{n}x_i}{n}\). Given \(n = 4\) and \(\bar{x}=92.0\), the sum of the original 4 numbers \(S_1=92\times4 = 368\).

Step2: Find the sum of the new 4 numbers

Let the sum of the first 3 numbers be \(x\). Then \(x+40=368\), so \(x = 328\). The sum of the new 4 numbers \(S_2=x + 48\). Substituting \(x = 328\), we get \(S_2=328+48=376\).

Step3: Calculate the average of the new list

Using the average formula \(\bar{y}=\frac{S_2}{4}\). Substituting \(S_2 = 376\), we get \(\bar{y}=\frac{376}{4}=94.0\).

Answer:

C. 18