Question

Three workers \(A, B\), and \(C\) are hired for 4 days. The daily wages of the three workers are as follows: A's first day wage is \(\$ 4\). Each day, his wage increases by 2 dollars. B's first day wage is \(\$ 3\). Each day, his wage increases by 2 dollars. C's first day wage is \(\$ 1\). Each day, his wage increases by the prime numbers 2,3 , and 5 in that order. Which one of the following is true about the wages earned by \(A, B\), and \(C\) in the first 4 days? (A) \(A>B>C\) (B) \(C>B>A\) (C) \(A>C>B\) (D) \(B>A>C\) (E) \(\quad C>A>B\)

Short answer

Option (A) is true: A > B > C.

Step-by-step solution

Calculating Total Wages of Worker A

Worker A's wage increases by $2 each day starting from the first day's wage of $4. We calculate this as follows: - Day 1: $4 - Day 2: $4 + $2 = $6 - Day 3: $6 + $2 = $8 - Day 4: $8 + $2 = $10 Total wages for A over 4 days = $4 + $6 + $8 + $10 = $28.

Calculating Total Wages of Worker B

Worker B's wage also increases by $2 each day starting from the first day's wage of $3. We calculate this as follows: - Day 1: $3 - Day 2: $3 + $2 = $5 - Day 3: $5 + $2 = $7 - Day 4: $7 + $2 = $9 Total wages for B over 4 days = $3 + $5 + $7 + $9 = $24.

Calculating Total Wages of Worker C

Worker C's wage increases by different prime numbers each day starting from the first day's wage of $1. The prime numbers are used in the order 2, 3, 5. We calculate this as follows: - Day 1: $1 - Day 2: $1 + $2 = $3 - Day 3: $3 + $3 = $6 - Day 4: $6 + $5 = $11 Total wages for C over 4 days = $1 + $3 + $6 + $11 = $21.

Comparing Total Wages of A, B, and C

Now, we compare the total wages: - Total wages for A = $28 - Total wages for B = $24 - Total wages for C = $21 The order of wages is A > B > C.

Key concepts

Arithmetic Sequences

An arithmetic sequence is a list of numbers where each term after the first is found by adding a constant to the previous term. This constant is known as the common difference. In our example with workers A and B, their wage increases form an arithmetic sequence. Let's break this down further:

• For Worker A, the first day's wage is \(4, and each day the wage increases by \)2. This makes an arithmetic sequence of 4, 6, 8, 10.
• Similarly, for Worker B, the initial wage is \(3, increasing by \)2 each day resulting in the sequence 3, 5, 7, 9.

To find any term in an arithmetic sequence, you can use the formula \( a_n = a_1 + (n-1)d \), where:

• \( a_n \) is the nth term
• \( a_1 \) is the first term
• \( d \) is the common difference
• \( n \) is the term number

This approach helps in calculating the total wages over multiple days efficiently as shown in the example. It simplifies the process to find each day's wage and eventually the total sum.

Prime Numbers

Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves. They are fundamental in number theory due to their role as 'building blocks' of the natural numbers. In this exercise, Worker C's wage increases are determined by prime numbers: 2, 3, and 5.

• Day 1 starts with a base wage of $1.
• On Day 2, 2 (a prime number) is added, resulting in $3.
• Day 3 adds 3, the next prime number, leading to $6.
• Finally, Day 4 sees an addition of 5, giving $11.

This unique pattern of wage calculation linked with prime numbers shows how primes can be used in practical arithmetic beyond their basic properties. While less predictable than arithmetic sequences, this pattern adds variety and complexity to wage modeling.

Wage Calculation

Wage calculation is a critical process in many real-world scenarios such as employment and project management. In this problem, understanding how the wages grow over a sequence of days is essential. Let's focus on the steps you can follow: 1. **Identify the Starting Wage:** Begin by identifying what each worker earns on the first day. 2. **Determine the Pattern:** Understand the pattern of wage increase. For example, Workers A and B have arithmetic increases, while Worker C's pattern involves increasing by different prime numbers. 3. **Calculate Daily Wages:** List out each day's wage by applying the increase pattern.

• For Worker A and B: Simply add a constant amount each day.
• For Worker C: Add a progressively larger prime number.

4. **Sum Up the Wages:** Add the daily wages to get the total amount for each worker over the provided period. This approach of methodically breaking down wage calculations helps in ensuring accuracy and clarity in financial assessments.