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What is Insertion of Rational Numbers Between Two Given Numbers?
Grade Level:
Class 7
AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering
Definition
What is it?
Insertion of rational numbers between two given numbers means finding other rational numbers that lie exactly in between them. Since there are infinitely many rational numbers between any two given rational numbers, you can always find more and more.
Simple Example
Quick Example
Imagine you have two friends, Rahul and Priya, who scored 70 and 80 marks in a Maths test. If you want to find a score between them, you could say 75. But you could also find 70.5, 70.75, or even 79.9. All these are rational numbers that lie between 70 and 80.
Worked Example
Step-by-Step
Find three rational numbers between 1/3 and 2/3.
Step 1: Make the denominators the same (they already are, which is 3). If they were different, we would find the LCM.
---Step 2: Multiply the numerator and denominator of both fractions by a number larger than 1 to create space. Let's multiply by 4.
1/3 = (1 * 4) / (3 * 4) = 4/12
2/3 = (2 * 4) / (3 * 4) = 8/12
---Step 3: Now, identify the rational numbers between 4/12 and 8/12.
These are 5/12, 6/12, and 7/12.
---Step 4: Simplify the fractions if possible.
5/12 (cannot be simplified)
6/12 = 1/2
7/12 (cannot be simplified)
Answer: Three rational numbers between 1/3 and 2/3 are 5/12, 1/2, and 7/12.
Why It Matters
This concept helps in understanding how numbers are distributed, which is crucial in fields like Data Science for data analysis and in Computer Science for programming algorithms. Engineers use this to find precise values within a range, ensuring accuracy in designs and calculations.
Common Mistakes
MISTAKE: Not making the denominators common before finding numbers. | CORRECTION: Always find a common denominator (LCM) for both fractions first to accurately compare and find numbers in between.
MISTAKE: Forgetting to multiply both numerator and denominator by the same number. | CORRECTION: To keep the fraction's value unchanged, whatever you multiply the numerator by, you must also multiply the denominator by.
MISTAKE: Stopping after finding just one number when asked for multiple. | CORRECTION: Remember there are infinitely many rational numbers. If you need more 'space' to find numbers, multiply the numerator and denominator by a larger number.
Practice Questions
Try It Yourself
QUESTION: Find one rational number between 1/2 and 3/4. | ANSWER: 5/8
QUESTION: Find two rational numbers between -1 and 1/2. | ANSWER: 0, 1/4 (or -1/2, -1/4, etc.)
QUESTION: Find three rational numbers between 0.2 and 0.3. Express them as fractions. | ANSWER: 21/100, 22/100 (or 11/50), 23/100
MCQ
Quick Quiz
Which of the following rational numbers lies between 1/5 and 2/5?
1/10
3/10
4/5
1/2
The Correct Answer Is:
B
To find a number between 1/5 and 2/5, we can write them as 2/10 and 4/10. 3/10 clearly lies between 2/10 and 4/10. Other options are outside this range.
Real World Connection
In the Real World
When a weather app like AccuWeather or Google Weather predicts a temperature range, say between 25°C and 30°C, meteorologists use rational numbers to pinpoint more precise values like 26.5°C or 28.75°C for hourly forecasts. This helps us plan our day, whether to carry a light jacket or just an umbrella.
Key Vocabulary
Key Terms
RATIONAL NUMBER: A number that can be expressed as a fraction p/q, where p and q are integers and q is not zero. | DENOMINATOR: The bottom part of a fraction, showing how many equal parts the whole is divided into. | NUMERATOR: The top part of a fraction, showing how many parts are being considered. | LCM (Least Common Multiple): The smallest positive integer that is a multiple of two or more given integers.
What's Next
What to Learn Next
Now that you can insert rational numbers, you're ready to explore operations like addition and subtraction of rational numbers. This will help you solve more complex problems involving fractions and decimals, which are everywhere in daily life!
