In this sequence of numbers, there is an important pattern to recognize. Each number is less than the previous number by a difference that is double the margin between the previous two numbers. For example, the difference between \(\displaystyle 19\) and \(\displaystyle 18\) is \(\displaystyle 1.\) And, the difference between \(\displaystyle 18\) and \(\displaystyle 16\) is \(\displaystyle 2\). Thus, the missing number must be less than \(\displaystyle 16\) by a difference of \(\displaystyle 4\)--which means that \(\displaystyle 12\) is the missing number. 

In this sequence of numbers, there is an important pattern to recognize. Each number is greater than the number prior to it by a margin of \(\displaystyle 75.\) Thus, \(\displaystyle 275 + 75 = 350\)

In this sequence of numbers, there is an important pattern to recognize. Each number is less than the previous number in the sequence by a difference of \(\displaystyle 7\). Thus, \(\displaystyle 52 - 7 = 45\).

In this sequence of numbers, there is an important pattern to recognize. Each number is greater than the number prior to it by a margin of \(\displaystyle 6\). In other words, each number in this list is a multiple of \(\displaystyle 6\). \(\displaystyle 6 \cdot 3= 18\), \(\displaystyle 6 \cdot 4= 24\), thus the next number in this sequence is the product of \(\displaystyle 6 \cdot 5\) which equals \(\displaystyle 30\).

In this sequence, every subsequent number is \(\displaystyle 7\) less than the preceding number. Given that the number that precedes \(\displaystyle w\) is \(\displaystyle 71\), the value of w is \(\displaystyle 71-7=64\). Therefore, \(\displaystyle 64\) is the correct answer. 

The relationship between the values is \(\displaystyle y=0.75x\),

where \(\displaystyle y\) represents the cost of purchased books and \(\displaystyle x\) represents the number of books purchased. 

Once we realize this, we can determine how much purchasing two books \(\displaystyle (x=2)\) would cost: 

\(\displaystyle y=0.75(2)=1.50\)

We can determine the relationship between the values by creating a ratio of number of cupcakes to cost:

\(\displaystyle \frac{3}{2.55}=\frac{1}{x}\) 

Where \(\displaystyle x\) represents the cost of 1 cupcake. 

We can now solve for \(\displaystyle x\): 

\(\displaystyle (3)(x)=(2.55)(1)\)

\(\displaystyle x=0.85\)

The cost of one cupcake is then $0.85

We were told the grades are from \(\displaystyle 30\) students. The most direct way to solve this problem is to add the numbers of students listed so far:

\(\displaystyle 14 + 6 + 2 + 0 =22\)

We know there is a total of \(\displaystyle 30\) students so we can set up the equation:

\(\displaystyle 22+B=30\) which leaves us with \(\displaystyle B=8\). 

So the missing value is eight.

We can determine the relationship between the values by creating a ratio of number of cupcakes to cost:

\(\displaystyle \frac{3}{2.55}=\frac{1}{x}\) 

Where \(\displaystyle x\) represents the cost of 1 cupcake. 

We can now solve for \(\displaystyle x\): 

\(\displaystyle (3)(x)=(2.55)(1)\)

\(\displaystyle x=0.85\)

The cost of one cupcake is then \(\displaystyle \$0.85\)

The blue circle of the Venn diagram depicts the number of students who prefer TV, the orange circle depicts the number of students who prefer radio, and the region of overlap indicates the number of students who like both. Therefore, 7 students like both TV and radio.