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To determine how many pairs of whole numbers [tex]\((a, b)\)[/tex] in the range from 1 to 9 form the ratio [tex]\(2:3\)[/tex], it is essential to recall what it means for two ratios to be equivalent. Specifically, the ratio [tex]\(a:b\)[/tex] is equivalent to [tex]\(2:3\)[/tex] if the following proportion holds:
[tex]\[ \frac{a}{b} = \frac{2}{3} \][/tex]
This equation can be rewritten as:
[tex]\[ a \cdot 3 = b \cdot 2 \][/tex]
Now we need to find all pairs [tex]\((a, b)\)[/tex] such that when [tex]\(a\)[/tex] is multiplied by 3, the result is the same as [tex]\(b\)[/tex] multiplied by 2, within the range of 1 to 9 for both [tex]\(a\)[/tex] and [tex]\(b\)[/tex].
To solve this, we will check each pair [tex]\((a, b)\)[/tex] independently and count those that satisfy the above relationship. Let's go through each value of [tex]\(a\)[/tex] (from 1 to 9) and find corresponding values of [tex]\(b\)[/tex] that maintain the given ratio.
1. [tex]\(a = 1\)[/tex]:
[tex]\[ 1 \cdot 3 = 3 \quad \text{so} \quad b = \frac{3}{2} \quad \text{(not an integer, discard)} \][/tex]
2. [tex]\(a = 2\)[/tex]:
[tex]\[ 2 \cdot 3 = 6 \quad \text{so} \quad b = 3 \][/tex]
The pair [tex]\((2, 3)\)[/tex] satisfies the condition.
3. [tex]\(a = 3\)[/tex]:
[tex]\[ 3 \cdot 3 = 9 \quad \text{so} \quad b = \frac{9}{2} \quad \text{(not an integer, discard)} \][/tex]
4. [tex]\(a = 4\)[/tex]:
[tex]\[ 4 \cdot 3 = 12 \quad b = 6 \quad \text{(out of range, discard)} \][/tex]
5. [tex]\(a = 5\)[/tex]:
[tex]\[ 5 \cdot 3 = 15 \quad b = 7.5 \quad \text{(not an integer, discard)} \][/tex]
6. [tex]\(a = 6\)[/tex]:
[tex]\[ 6 \cdot 3 = 18 \quad b = 9 \][/tex]
The pair [tex]\((6, 9)\)[/tex] satisfies the condition.
7. [tex]\(a = 7\)[/tex]:
[tex]\[ 7 \cdot 3 = 21 \quad b = 10.5 \quad \text{(not an integer, discard)} \][/tex]
8. [tex]\(a = 8\)[/tex]:
[tex]\[ 8 \cdot 3 = 24 \quad b = 12 \quad \text{(out of range, discard)} \][/tex]
9. [tex]\(a = 9\)[/tex]:
[tex]\[ 9 \cdot 3 = 27 \quad b = 13.5 \quad \text{(not an integer, discard)} \][/tex]
From this thorough inspection, we can see three pairs of numbers within the range [tex]\(1\)[/tex] through [tex]\(9\)[/tex] that satisfy the ratio [tex]\(2:3\)[/tex]:
- [tex]\((2, 3)\)[/tex]
- [tex]\((4, 6)\)[/tex]
- [tex]\((6, 9)\)[/tex]
Therefore, there are 3 such pairs, and the number of ratios equivalent to [tex]\(2:3\)[/tex] using whole numbers from 1 to 9 is:
[tex]\[ \boxed{3} \][/tex]
[tex]\[ \frac{a}{b} = \frac{2}{3} \][/tex]
This equation can be rewritten as:
[tex]\[ a \cdot 3 = b \cdot 2 \][/tex]
Now we need to find all pairs [tex]\((a, b)\)[/tex] such that when [tex]\(a\)[/tex] is multiplied by 3, the result is the same as [tex]\(b\)[/tex] multiplied by 2, within the range of 1 to 9 for both [tex]\(a\)[/tex] and [tex]\(b\)[/tex].
To solve this, we will check each pair [tex]\((a, b)\)[/tex] independently and count those that satisfy the above relationship. Let's go through each value of [tex]\(a\)[/tex] (from 1 to 9) and find corresponding values of [tex]\(b\)[/tex] that maintain the given ratio.
1. [tex]\(a = 1\)[/tex]:
[tex]\[ 1 \cdot 3 = 3 \quad \text{so} \quad b = \frac{3}{2} \quad \text{(not an integer, discard)} \][/tex]
2. [tex]\(a = 2\)[/tex]:
[tex]\[ 2 \cdot 3 = 6 \quad \text{so} \quad b = 3 \][/tex]
The pair [tex]\((2, 3)\)[/tex] satisfies the condition.
3. [tex]\(a = 3\)[/tex]:
[tex]\[ 3 \cdot 3 = 9 \quad \text{so} \quad b = \frac{9}{2} \quad \text{(not an integer, discard)} \][/tex]
4. [tex]\(a = 4\)[/tex]:
[tex]\[ 4 \cdot 3 = 12 \quad b = 6 \quad \text{(out of range, discard)} \][/tex]
5. [tex]\(a = 5\)[/tex]:
[tex]\[ 5 \cdot 3 = 15 \quad b = 7.5 \quad \text{(not an integer, discard)} \][/tex]
6. [tex]\(a = 6\)[/tex]:
[tex]\[ 6 \cdot 3 = 18 \quad b = 9 \][/tex]
The pair [tex]\((6, 9)\)[/tex] satisfies the condition.
7. [tex]\(a = 7\)[/tex]:
[tex]\[ 7 \cdot 3 = 21 \quad b = 10.5 \quad \text{(not an integer, discard)} \][/tex]
8. [tex]\(a = 8\)[/tex]:
[tex]\[ 8 \cdot 3 = 24 \quad b = 12 \quad \text{(out of range, discard)} \][/tex]
9. [tex]\(a = 9\)[/tex]:
[tex]\[ 9 \cdot 3 = 27 \quad b = 13.5 \quad \text{(not an integer, discard)} \][/tex]
From this thorough inspection, we can see three pairs of numbers within the range [tex]\(1\)[/tex] through [tex]\(9\)[/tex] that satisfy the ratio [tex]\(2:3\)[/tex]:
- [tex]\((2, 3)\)[/tex]
- [tex]\((4, 6)\)[/tex]
- [tex]\((6, 9)\)[/tex]
Therefore, there are 3 such pairs, and the number of ratios equivalent to [tex]\(2:3\)[/tex] using whole numbers from 1 to 9 is:
[tex]\[ \boxed{3} \][/tex]
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