LET GENERAL EDUCATION REVIEWER
INVERSE PROPORTION
Workers and Days | Practice Question | Answer and Rationale
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Introduction
Mathematics is an important component of the General Education portion of the Licensure Examination for Professional Teachers (LET/LEPT). Examinees must understand mathematical relationships and develop problem-solving skills to answer different types of word problems.
One important topic in Mathematics is inverse proportion. This concept is commonly applied to problems involving workers and time, machines and production, speed and travel time, and other situations where one quantity decreases as another increases.
In this LET practice question, we will learn how to calculate the number of workers required to finish a particular job within a shorter period using the concept of inverse proportion.
Table of Contents
LET Practice Question
Question:
To finish a certain job in 6 days, 8 workers are needed. If it is required to finish the same job 2 days earlier, how many workers have to work?
A. 12
B. 10
C. 6
D. 14
Correct Answer
A. 12 workers
Twelve workers are needed to finish the same job in 4 days, assuming all workers have equal productivity.
Detailed Rationale
To solve this problem, we need to understand the relationship between the number of workers and the number of days required to complete a fixed amount of work.
Given:
- 8 workers are needed.
- The original completion time is 6 days.
- The job must be completed 2 days earlier.
Step 1: Determine the required number of days.
The job must be completed 2 days earlier than the original 6-day schedule.
6 days - 2 days = 4 days
Therefore, the required completion time is 4 days.
Step 2: Identify the mathematical relationship.
Since the amount of work remains the same, the number of workers and the number of days are inversely proportional, assuming equal working efficiency.
We use the formula:
Workers × Days = Constant Work
8 × 6 = x × 4
Step 3: Solve for the unknown number of workers.
First, multiply the original number of workers by the original number of days.
8 × 6 = 48 worker-days
Next, divide the total worker-days by the required number of days.
x = 48 ÷ 4
x = 12 workers
FINAL ANSWER: 12 WORKERS
The correct choice is Letter A.
Analysis of All Choices
Option A: 12 – CORRECT
Twelve workers are needed to finish the job in 4 days. The total work remains constant at 48 worker-days.
12 × 4 = 48 worker-days.
Option B: 10 – INCORRECT
Ten workers would require 4.8 days to complete the job under the same working conditions.
10 × 4.8 = 48 worker-days.
This is longer than the required 4 days.
Option C: 6 – INCORRECT
Six workers would require 8 days to complete the same job.
6 × 8 = 48 worker-days.
Reducing the number of workers increases the time needed to finish the job. Therefore, 6 workers cannot complete the task within the required 4 days.
Option D: 14 – INCORRECT
Fourteen workers would require approximately 3.43 days to finish the job.
Although this is sufficient to finish within 4 days, it is more workers than necessary. The question asks for the number of workers needed to complete the job in exactly 4 days under the given assumptions.
Key Mathematical Concepts to Remember
1. Inverse Proportion
Inverse proportion occurs when two quantities change in opposite directions while their product remains constant.
For example, when the number of workers increases, the number of days required to complete a fixed amount of work decreases, provided that the workers have equal productivity.
2. Direct Proportion
Direct proportion occurs when two quantities increase or decrease together at a constant rate.
For example, if one notebook costs P20, buying more notebooks at the same price per notebook increases the total cost proportionally.
3. Worker-Days
Worker-days represent the amount of work completed by a given number of workers over a specified number of days, assuming consistent working hours and productivity.
The formula is:
Total Work = Number of Workers × Number of Days
This formula is useful for solving problems involving workers, time, and a fixed amount of work.
4. Important Assumptions
These problems generally assume that:
- All workers have equal productivity.
- Each worker works the same number of hours per day.
- The amount of work remains constant.
- There are no significant interruptions or changes in working conditions.
Real-Life Application
Suppose four workers can paint a classroom in eight days. The school administration wants the same painting job completed in four days.
How many workers are needed?
Step 1: Calculate the total worker-days.
4 × 8 = 32 worker-days.
Step 2: Divide by the required completion time.
32 ÷ 4 = 8 workers.
Answer: 8 workers.
This illustrates inverse proportion: reducing the completion time by half requires twice as many workers, assuming equal productivity.
LET Examination Tips
When answering Mathematics questions involving work problems, follow these strategies:
- Identify the original time. Determine how many days are initially required.
- Calculate the new time. Pay attention to expressions such as earlier, later, faster, or slower.
- Identify the relationship. For a fixed amount of work and equal productivity, workers and days are inversely proportional.
- Use the worker-days formula. Multiply workers by days to obtain the total work.
- Check your answer. More workers should generally mean fewer days for the same job.
Memory Aid
INVERSE PROPORTION:
More workers = Fewer days.
Fewer workers = More days.
Remember the formula:
Workers × Days = Total Work
For this problem:
8 × 6 = 12 × 4
48 = 48
Frequently Asked Questions
1. What is inverse proportion in Mathematics?
Inverse proportion is a relationship between two quantities in which an increase in one quantity results in a decrease in the other, while their product remains constant.
2. How do you solve workers and days problems?
Multiply the original number of workers by the original number of days to determine the total worker-days. Divide the result by the required number of days to calculate the new number of workers.
3. What is the difference between direct and inverse proportion?
In direct proportion, two quantities increase or decrease together. In inverse proportion, one quantity increases while the other decreases, assuming the relationship remains constant.
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Final Encouragement for Future LPTs
Every Mathematics problem you solve strengthens your ability to think logically and make sound decisions. Do not be afraid of word problems. Identify the given information, understand the mathematical relationship, and apply the correct formula.
Continue practicing and developing your problem-solving skills. Your journey toward becoming a licensed professional teacher is worth every effort.
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