Sure, let's go step-by-step to understand and illustrate the equivalence of the fractions [tex]\(\frac{3}{4}\)[/tex] and [tex]\(\frac{9}{12}\)[/tex] using a number line.
### Step 1: Understand the Fractions
- The fraction [tex]\(\frac{3}{4}\)[/tex] means 3 parts out of 4 equal parts.
- The fraction [tex]\(\frac{9}{12}\)[/tex] means 9 parts out of 12 equal parts.
### Step 2: Simplify the Fractions
To check if the fractions are equivalent, we simplify [tex]\(\frac{9}{12}\)[/tex]:
- Divide both the numerator (9) and the denominator (12) by their greatest common divisor, which is 3.
- [tex]\(\frac{9 \div 3}{12 \div 3} = \frac{3}{4}\)[/tex].
Since [tex]\(\frac{9}{12}\)[/tex] simplifies to [tex]\(\frac{3}{4}\)[/tex], we verify that they are indeed equivalent.
### Step 3: Place [tex]\(\frac{3}{4}\)[/tex] on the Number Line
1. Draw a number line and mark the integers 0 and 1.
2. Divide the segment from 0 to 1 into 4 equal parts because the denominator of [tex]\(\frac{3}{4}\)[/tex] is 4.
3. The first division represents [tex]\(\frac{1}{4}\)[/tex], the second represents [tex]\(\frac{2}{4}\)[/tex] (or [tex]\(\frac{1}{2}\)[/tex]), the third represents [tex]\(\frac{3}{4}\)[/tex].
4. Place a point at the 3rd division to represent [tex]\(\frac{3}{4}\)[/tex].
### Step 4: Place [tex]\(\frac{9}{12}\)[/tex] on the Number Line
Since [tex]\(\frac{9}{12}\)[/tex] is equivalent to [tex]\(\frac{3}{4}\)[/tex], it should fall at the same point as [tex]\(\frac{3}{4}\)[/tex].
1. If you were to divide the segment from 0 to 1 into 12 equal parts for [tex]\(\frac{9}{12}\)[/tex], each part represents [tex]\(\frac{1}{12}\)[/tex].
2. The first division represents [tex]\(\frac{1}{12}\)[/tex], the second [tex]\(\frac{2}{12}\)[/tex], and so on.
3. The ninth division represents [tex]\(\frac{9}{12}\)[/tex].
4. Place a point at the 9th division – it should coincide exactly with the point where [tex]\(\frac{3}{4}\)[/tex] is placed.
### Step 5: Verify Equivalence
Both points for [tex]\(\frac{3}{4}\)[/tex] and [tex]\(\frac{9}{12}\)[/tex] on the number line coincide, and therefore, we can see that they are equivalent fractions.
By following these detailed steps, we clearly see that [tex]\(\frac{3}{4}\)[/tex] and [tex]\(\frac{9}{12}\)[/tex] are indeed equivalent, as they represent the same point on the number line.
