Problem 51 - Algebra 2

Question

Kono’s employer gives him 1.5 vacation days for each month he works. If Kono has 11 days at the end of one year and takes no vacation time during the next year, how many days will he have at the end of that year?

Step-by-Step Solution

Answer

Kono will have 36 vacation days at the end of the second year.

Step 1: Determine the number of vacation days earned in a year

Kono earns 1.5 vacation days for each month he works. There are 12 months in a year, so we need to calculate how many vacation days he earns in one year: \(1.5 \text{ days/month} \times 12 \text{ months} = 18 \text{ days}\).

Step 2: Calculate the total vacation days at the end of the first year

At the end of the first year, Kono has used no vacation days and has earned 18 days through his employer's policy, so he has 18 vacation days available.

Step 3: Calculate the total number of vacation days at the end of the second year

At the start of the second year, Kono has 18 vacation days from the first year. Over the course of the second year, he earns another 18 days: \(18 \text{ days (from first year)} + 18 \text{ days (earned in second year)} = 36 \text{ days}\).

Key Concepts

Arithmetic Operations

Arithmetic operations are fundamental mathematical actions that involve adding, subtracting, multiplying, and dividing numbers. These operations form the backbone of everyday calculations, allowing us to understand and interpret various numerical data. In the context of calculating Kono's vacation days, arithmetic operations help us add and multiply to determine how many days he accumulates over time.

• Addition: Used to combine quantities. For example, when we add vacation days earned from the first and second year.
• Multiplication: Allows us to find the total number of something repeated several times. In Kono's case, multiplying the number of days earned per month by the number of months in a year gives us his annual vacation days.

Understanding these operations aids in solving real-world problems involving calculations, just like how Kono calculates his number of vacation days.

Multiplication in Mathematics

Multiplication is a mathematical operation where a number is added to itself a certain number of times. It is represented by the symbol \( \times \). In Kono's scenario, multiplication helps us compute how many vacation days he accumulates throughout the year based on what he earns monthly. For Kono's vacation day calculation:

• The formula is given by: \( 1.5 \times 12 = 18 \), showing that if Kono earns 1.5 days each month, multiplying by 12 months results in 18 days.
• This reflects how multiplication translates a monthly earning into a yearly total unambiguously, crafting a straightforward model for calculating repeated additions efficiently.

Using multiplication provides accuracy and efficiency in calculations, making it an essential tool in both academic and real-life scenarios.

Real-life Math Application

Real-life math application pertains to using mathematical concepts to solve everyday problems. Calculating vacation days, like in Kono's situation, is a practical example of applying mathematics in real life. Here's how math concepts assist in understanding and managing vacation days:

• Understanding Earned Benefits: By knowing how to calculate the days he should earn, Kono can plan his vacations better and ensure he's being compensated correctly.
• Household Budgeting: Estimating how many days off one has can aid in budgeting time and expenses for vacation plans.
• Work-life Balance Planning: Math helps Kono keep track of his authorized leave, allowing for balanced work-life schedules without disruptions.

This exercise shows that math is not just about numbers on a page but a vital skill for organizing and enhancing daily life, making calculations like vacation days seamless and systematic.