Question

A machine manufactures notebooks in a series of five colors: red, blue, black, white, and yellow. After producing a notebook of one color from that series, it produces a notebook of the next color. Once five are produced, the machine repeats the same pattern. If the machine began a day producing a red notebook and completed the day by producing a black notebook, how many notebooks could have been produced that day? A. $$27$$ B. $$34$$ C. $$50$$ D. $$61$$ E. $$78$$

Short answer

E. 78

Step-by-step solution

Identify the sequence of colors

The machine produces notebooks in the following order: red, blue, black, white, yellow. This sequence repeats every five notebooks.

Determine the color pattern

Given that the day starts with a red notebook and ends with a black notebook, identify the possible range within which this cycle might occur.

Define progression within one cycle

In a single cycle, the sequence is red, blue, black, white, yellow. The third notebook in this cycle is black.

Consider multiple cycles

Since the day starts with red and ends with black, it must complete several full cycles (multiple sets of 5 notebooks) before stopping at black in a subsequent cycle.

Set up the equation for the total number of notebooks

Let the number of full 5-notebook cycles be denoted by \(n\). Each cycle produces 5 notebooks. If the machine completes a full cycle \(n\) times, it produces \(5n\) notebooks. After these cycles, the notebook sequence started with red, going through blue (2nd), and ends with black (3rd). Hence, the total number of notebooks is given by: Formula: Total notebooks = \(5n + 3\)

Calculate possible values and verify

To find the value of \(n\), test the options: A: 27 = \(5n + 3\) → \(n = 24/5 → n = 4.8\) (invalid, not an integer) B: 34 = \(5n + 3\) → \(n = 31/5 → n = 6.2\) (invalid, not an integer) C: 50 = \(5n + 3\) → \(n = 47/5 → n = 9.4\) (invalid, not an integer) D: 61 = \(5n + 3\) → \(n = 58/5 → n = 11.6\) (invalid, not an integer) E: 78 = \(5n + 3\) → \(n = 75/5 → n = 15\) (valid integer) Thus, the option E (78) satisfies the condition and is correct.

Key concepts

Pattern Recognition

Pattern recognition is the ability to spot repeated designs or recurring events in a sequence. In the given notebook manufacturing exercise, the machine produces notebooks in a predictable color pattern: red, blue, black, white, and yellow. By identifying this pattern, we simplify solving the problem.
Recognizing patterns lets us predict future outputs, including which color a notebook will be after a given number of cycles. To get better at pattern recognition:

• Look for sequences that repeat.
• Keep track of each element in the sequence.
• Calculate how many elements make up one full cycle.
• Practice with different types of sequences to improve your skills.

Cyclic Sequences

Cyclic sequences are patterns that repeat after a certain number of elements. In this problem, the machine's production sequence cycles every five notebooks. After producing five different colored notebooks, it starts the pattern again.
Understanding cyclic sequences can help in determining points where the cycle restarts. This makes it easier to figure out larger sequences without listing every element. For this notebook example:

• The cycle contains five elements: red, blue, black, white, yellow.
• The sequence restarts after every five notebooks.
• The formula to find the position in the cycle after otebooks is given by: otebooks % cycle_length.

Algebraic Equations

Algebraic equations are used to model relationships and solve for unknown variables. In our notebook problem, we set up an equation to represent the total notebooks produced. By defining the number of full cycles as , we get notebooks = 5n.
Since the machine ends with a black notebook, a third element in the next cycle, we add 3 more notebooks: Total notebooks = 5n + 3.
Using this equation, we tested different options to find how many notebooks were produced. The goal is to find such that the total number of notebooks is a whole number.
Steps to solve algebraic equations:

• Identify the variables and constants.
• Write the equation representing the problem.
• Plug in the given values and solve for the variable.
• Check if the solution fits the problem's conditions.

Integer Validation

Integer validation ensures the solutions we find are whole numbers since in many real-world problems, fractional values don't make sense. For our notebook problem, the total number of notebooks must be an integer.
During our calculations, we tested each option's validity by ensuring is a whole number. This meant that:
27, 34, 50, and 61 were not feasible since their -values weren't integers.
However, for 78, =15, which is a valid integer.
Steps for integer validation:

• Solve the equation with the given value.
• Check if the solution is an integer by evaluating the fractional part.
• Conclude if the solution is valid or not based on the presence/absence of a fractional part.