Equivalent Rational Numbers
In fractions, you learned that 1/2, 2/4, and 3/6 are all the same — they are equivalent fractions. The same idea applies to rational numbers, which include negative fractions as well.
Rational numbers are numbers that can be written in the form p/q, where p and q are integers and q ≠ 0. Two rational numbers are equivalent if they represent the same value, even though they look different.
For example, 2/3 and 4/6 are equivalent rational numbers. So are −3/5 and 6/(−10). Finding equivalent rational numbers is essential for comparing, adding, and subtracting rational numbers.
In NCERT Class 7, this topic appears in the chapter Rational Numbers. It builds on what you know about equivalent fractions and extends it to include negative numbers.
What is Equivalent Rational Numbers?
Definition: Two rational numbers are called equivalent rational numbers if they represent the same value.
How to check equivalence:
- Two rational numbers a/b and c/d are equivalent if a × d = b × c (cross-multiplication gives equal products).
- Alternatively, if you can multiply or divide both the numerator and denominator of one rational number by the same non-zero integer to get the other, they are equivalent.
Key points:
- Multiplying the numerator and denominator by the same non-zero integer gives an equivalent rational number.
- Dividing the numerator and denominator by their common factor gives an equivalent rational number.
- Multiplying or dividing by a negative number is also valid — it changes signs of both numerator and denominator.
Important:
- Every rational number has infinitely many equivalent forms.
- The simplest form (or standard form) is the one where the numerator and denominator have no common factor other than 1, and the denominator is positive.
Equivalent Rational Numbers Formula
Rule for generating equivalent rational numbers:
a/b = (a × m) / (b × m)
Where m is any non-zero integer (positive or negative).
Cross-multiplication test:
a/b = c/d if and only if a × d = b × c
Reducing to simplest form:
- Divide both numerator and denominator by their HCF.
- Make sure the denominator is positive.
Examples of equivalent rational numbers:
- 1/2 = 2/4 = 3/6 = 4/8 = −1/(−2) = −2/(−4)
- −3/4 = −6/8 = −9/12 = 3/(−4) = 6/(−8)
Types and Properties
Types of problems on equivalent rational numbers:
1. Generating equivalent rational numbers:
- Multiply numerator and denominator by 2, 3, 4, etc.
- Example: 3/5 → 6/10, 9/15, 12/20, ...
2. Checking if two rational numbers are equivalent:
- Use cross-multiplication: a × d = b × c?
- Example: Is 2/3 = 8/12? Check: 2 × 12 = 24 and 3 × 8 = 24. Yes, they are equal.
3. Reducing to simplest form:
- Find the HCF of numerator and denominator, then divide both by the HCF.
- Example: 12/18 → HCF = 6 → 2/3.
4. Finding a missing numerator or denominator:
- Given that a/b = ?/d, use cross-multiplication to find the missing number.
- Example: 3/7 = ?/21 → ? = 3 × 21/7 = 9.
5. Equivalent rational numbers with negative signs:
- −a/b = a/(−b) = −(a/b). The negative sign can be with numerator, denominator, or in front of the fraction.
Solved Examples
Example 1: Example 1: Generating equivalent rational numbers
Problem: Write four equivalent rational numbers for 3/5.
Solution:
Multiply numerator and denominator by 2, 3, 4, and 5:
- 3/5 = (3 × 2)/(5 × 2) = 6/10
- 3/5 = (3 × 3)/(5 × 3) = 9/15
- 3/5 = (3 × 4)/(5 × 4) = 12/20
- 3/5 = (3 × 5)/(5 × 5) = 15/25
Answer: Four equivalent rational numbers for 3/5 are 6/10, 9/15, 12/20, and 15/25.
Example 2: Example 2: Equivalent forms of a negative rational number
Problem: Write three equivalent rational numbers for −2/7.
Solution:
Multiply numerator and denominator by 2, 3, and −1:
- −2/7 = (−2 × 2)/(7 × 2) = −4/14
- −2/7 = (−2 × 3)/(7 × 3) = −6/21
- −2/7 = (−2 × −1)/(7 × −1) = 2/(−7)
Note: 2/(−7) = −2/7. The negative sign can be placed with the numerator, denominator, or in front of the fraction.
Answer: Three equivalent rational numbers are −4/14, −6/21, and 2/(−7).
Example 3: Example 3: Checking equivalence using cross-multiplication
Problem: Are 4/6 and 10/15 equivalent rational numbers?
Solution:
Using cross-multiplication:
- 4 × 15 = 60
- 6 × 10 = 60
- Since 60 = 60, the cross-products are equal.
Answer: Yes, 4/6 and 10/15 are equivalent. (Both simplify to 2/3.)
Example 4: Example 4: Checking non-equivalence
Problem: Are 3/8 and 5/12 equivalent?
Solution:
Using cross-multiplication:
- 3 × 12 = 36
- 8 × 5 = 40
- Since 36 ≠ 40, the cross-products are not equal.
Answer: No, 3/8 and 5/12 are not equivalent.
Example 5: Example 5: Reducing to simplest form
Problem: Reduce 36/48 to its simplest form.
Solution:
Step 1: Find the HCF of 36 and 48.
- Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
- Factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48
- HCF = 12
Step 2: Divide both by HCF:
- 36/48 = (36 ÷ 12) / (48 ÷ 12) = 3/4
Answer: The simplest form is 3/4.
Example 6: Example 6: Simplifying a negative rational number
Problem: Reduce −18/24 to its simplest form.
Solution:
Step 1: Find the HCF of 18 and 24.
- HCF(18, 24) = 6
Step 2: Divide both by HCF:
- −18/24 = (−18 ÷ 6) / (24 ÷ 6) = −3/4
Answer: The simplest form is −3/4.
Example 7: Example 7: Finding the missing number
Problem: Find the value of x: 5/9 = x/36.
Solution:
Using cross-multiplication:
- 5/9 = x/36
- 5 × 36 = 9 × x
- 180 = 9x
- x = 180/9 = 20
Verification: 5/9 = 20/36. Simplify 20/36: divide by 4 → 5/9 ✓
Answer: x = 20.
Example 8: Example 8: Missing value with negative rational number
Problem: Find the value of y: −4/5 = 12/y.
Solution:
Using cross-multiplication:
- −4/5 = 12/y
- −4 × y = 5 × 12
- −4y = 60
- y = 60/(−4) = −15
Verification: −4/5 = 12/(−15). Simplify 12/(−15): divide by 3 → 4/(−5) = −4/5 ✓
Answer: y = −15.
Example 9: Example 9: Checking three forms of a negative rational number
Problem: Show that −3/5, 3/(−5), and −(3/5) are all the same rational number.
Solution:
Method: The negative sign can be placed in three positions:
- −3/5: negative sign on the numerator
- 3/(−5): negative sign on the denominator
- −(3/5): negative sign in front of the fraction
Check using decimals:
- −3/5 = −0.6
- 3/(−5) = −0.6
- −(3/5) = −0.6
All three give the same value: −0.6.
Standard convention: The negative sign is usually written with the numerator or in front: −3/5.
Answer: All three are equivalent and equal to −3/5 (or −0.6).
Example 10: Example 10: Finding equivalent rational number with given denominator
Problem: Express −7/8 as a rational number with denominator 32.
Solution:
Step 1: Find the multiplier for the denominator:
- 32 ÷ 8 = 4
Step 2: Multiply both numerator and denominator by 4:
- −7/8 = (−7 × 4) / (8 × 4) = −28/32
Verification: −28/32 → divide by 4 → −7/8 ✓
Answer: −7/8 = −28/32.
Real-World Applications
Real-world uses of equivalent rational numbers:
- Cooking and recipes: When scaling recipes, you use equivalent fractions. Half a cup (1/2) is the same as 2/4 of a cup. If a recipe needs 3/4 cup of flour and you double it, you need 6/4 = 3/2 cups.
- Comparing prices: To compare prices of different pack sizes, you express them as equivalent fractions with the same denominator. This is like finding the "per unit" price.
- Adding and subtracting fractions: Before adding 1/3 and 1/4, you need to find equivalent fractions with the same denominator (4/12 and 3/12). This is the core use of equivalent rational numbers.
- Maps and scales: A map scale of 1:50000 means 1 cm on the map = 50,000 cm in reality. Equivalent ratios help convert between map and real distances.
- Music: Musical notes have frequency ratios. Equivalent ratios explain why certain note combinations sound harmonious.
- Probability: The probability of an event can be expressed in different equivalent forms: 1/4 = 2/8 = 25/100 = 25%.
Key Points to Remember
- Two rational numbers are equivalent if they represent the same value.
- To generate an equivalent rational number, multiply or divide both numerator and denominator by the same non-zero integer.
- Cross-multiplication test: a/b = c/d if and only if a × d = b × c.
- Every rational number has infinitely many equivalent forms.
- The simplest form is obtained by dividing numerator and denominator by their HCF.
- In the simplest form, the denominator must be positive.
- −a/b = a/(−b) = −(a/b) — the negative sign can be in any of these three positions.
- The standard convention is to write the negative sign with the numerator or in front of the fraction.
- Finding equivalent rational numbers is essential for comparing, adding, and subtracting rational numbers.
- 0 can be written as 0/1, 0/2, 0/3, etc. — all are equivalent forms of zero.
Practice Problems
- Write five equivalent rational numbers for 2/7.
- Write three equivalent rational numbers for −5/6.
- Check whether 4/9 and 16/36 are equivalent.
- Check whether −3/7 and 9/−21 are equivalent.
- Reduce 45/60 to its simplest form.
- Reduce −24/36 to its simplest form.
- Find the value of x if 3/11 = x/55.
- Express 5/6 as a rational number with denominator 42.
Frequently Asked Questions
Q1. What are equivalent rational numbers?
Equivalent rational numbers are rational numbers that have the same value even though they are written differently. For example, 1/2 and 3/6 are equivalent because they both equal 0.5. You get equivalent rational numbers by multiplying or dividing both the numerator and denominator by the same non-zero integer.
Q2. How do you check if two rational numbers are equivalent?
Use cross-multiplication. For a/b and c/d, if a × d = b × c, they are equivalent. For example, to check 2/3 and 8/12: 2 × 12 = 24 and 3 × 8 = 24. Since 24 = 24, they are equivalent.
Q3. How many equivalent rational numbers can a rational number have?
Infinitely many. You can keep multiplying the numerator and denominator by 2, 3, 4, 5, ... to get 2/4, 3/6, 4/8, ... (for 1/2). There is no end to this process.
Q4. What is the simplest form of a rational number?
The simplest form (or lowest terms) is when the numerator and denominator have no common factor other than 1, and the denominator is positive. For example, the simplest form of 6/8 is 3/4 (divide both by HCF = 2).
Q5. Is −3/5 the same as 3/(−5)?
Yes. −3/5 = 3/(−5) = −(3/5). The negative sign can be placed on the numerator, the denominator, or in front of the fraction — all three represent the same rational number. The standard way is to write it as −3/5.
Q6. How is this different from equivalent fractions?
Equivalent fractions deal with positive fractions (like 1/2 = 2/4). Equivalent rational numbers extend this to include negative numbers (like −1/2 = −2/4 = 1/(−2)). The process is the same — multiply or divide by the same non-zero integer — but rational numbers include negatives.
Q7. Is 0/5 equivalent to 0/3?
Yes. Any rational number with 0 as the numerator equals 0. So 0/5 = 0/3 = 0/100 = 0. They are all equivalent forms of zero.
Q8. Why is finding equivalent rational numbers useful?
It is essential for adding and subtracting rational numbers (you need a common denominator), for comparing rational numbers, and for simplifying answers. Without equivalent forms, you cannot perform these operations.
Related Topics
- Rational Numbers
- Standard Form of Rational Number
- Comparing Rational Numbers
- Equivalent Fractions
- Rational Numbers on Number Line
- Operations on Rational Numbers
- Properties of Rational Numbers
- Rational Numbers Between Two Numbers
- Additive Inverse of Rational Numbers
- Multiplicative Inverse of Rational Numbers
- Density Property of Rational Numbers
