Answer :
Sure! Let's determine whether each pair of ratios forms a proportion by using the cross-multiplication method. Cross-multiplication is a way to compare two fractions or ratios by multiplying the numerator of one ratio by the denominator of the other ratio. If the products are equal, the pairs form a proportion.
### Step-by-Step Solution:
a. Check if [tex]\(\frac{2}{6}\)[/tex] and [tex]\(\frac{8}{24}\)[/tex] form a proportion.
1. Perform cross-multiplication:
- Multiply 2 (numerator of the first ratio) by 24 (denominator of the second ratio).
- Multiply 6 (denominator of the first ratio) by 8 (numerator of the second ratio).
2. Calculate the products:
- [tex]\(2 \times 24 = 48\)[/tex]
- [tex]\(6 \times 8 = 48\)[/tex]
3. Compare the products: [tex]\(48 = 48\)[/tex]
Since the products are equal, [tex]\(\frac{2}{6}\)[/tex] and [tex]\(\frac{8}{24}\)[/tex] do form a proportion.
Result: True
---
b. Check if [tex]\(\frac{6}{10}\)[/tex] and [tex]\(\frac{9}{12}\)[/tex] form a proportion.
1. Perform cross-multiplication:
- Multiply 6 by 12.
- Multiply 10 by 9.
2. Calculate the products:
- [tex]\(6 \times 12 = 72\)[/tex]
- [tex]\(10 \times 9 = 90\)[/tex]
3. Compare the products: [tex]\(72 \neq 90\)[/tex]
Since the products are not equal, [tex]\(\frac{6}{10}\)[/tex] and [tex]\(\frac{9}{12}\)[/tex] do not form a proportion.
Result: False
---
c. Check if [tex]\(\frac{21}{30}\)[/tex] and [tex]\(\frac{35}{50}\)[/tex] form a proportion.
1. Perform cross-multiplication:
- Multiply 21 by 50.
- Multiply 30 by 35.
2. Calculate the products:
- [tex]\(21 \times 50 = 1050\)[/tex]
- [tex]\(30 \times 35 = 1050\)[/tex]
3. Compare the products: [tex]\(1050 = 1050\)[/tex]
Since the products are equal, [tex]\(\frac{21}{30}\)[/tex] and [tex]\(\frac{35}{50}\)[/tex] do form a proportion.
Result: True
---
d. Check if [tex]\(\frac{1.8}{7.5}\)[/tex] and [tex]\(\frac{1.2}{5}\)[/tex] form a proportion.
1. Perform cross-multiplication:
- Multiply 1.8 by 5.
- Multiply 7.5 by 1.2.
2. Calculate the products:
- [tex]\(1.8 \times 5 = 9.0\)[/tex]
- [tex]\(7.5 \times 1.2 = 9.0\)[/tex]
3. Compare the products: [tex]\(9.0 = 9.0\)[/tex]
Since the products are equal, [tex]\(\frac{1.8}{7.5}\)[/tex] and [tex]\(\frac{1.2}{5}\)[/tex] do form a proportion.
Result: True
I hope this helps explain whether each pair of ratios forms a proportion!
### Step-by-Step Solution:
a. Check if [tex]\(\frac{2}{6}\)[/tex] and [tex]\(\frac{8}{24}\)[/tex] form a proportion.
1. Perform cross-multiplication:
- Multiply 2 (numerator of the first ratio) by 24 (denominator of the second ratio).
- Multiply 6 (denominator of the first ratio) by 8 (numerator of the second ratio).
2. Calculate the products:
- [tex]\(2 \times 24 = 48\)[/tex]
- [tex]\(6 \times 8 = 48\)[/tex]
3. Compare the products: [tex]\(48 = 48\)[/tex]
Since the products are equal, [tex]\(\frac{2}{6}\)[/tex] and [tex]\(\frac{8}{24}\)[/tex] do form a proportion.
Result: True
---
b. Check if [tex]\(\frac{6}{10}\)[/tex] and [tex]\(\frac{9}{12}\)[/tex] form a proportion.
1. Perform cross-multiplication:
- Multiply 6 by 12.
- Multiply 10 by 9.
2. Calculate the products:
- [tex]\(6 \times 12 = 72\)[/tex]
- [tex]\(10 \times 9 = 90\)[/tex]
3. Compare the products: [tex]\(72 \neq 90\)[/tex]
Since the products are not equal, [tex]\(\frac{6}{10}\)[/tex] and [tex]\(\frac{9}{12}\)[/tex] do not form a proportion.
Result: False
---
c. Check if [tex]\(\frac{21}{30}\)[/tex] and [tex]\(\frac{35}{50}\)[/tex] form a proportion.
1. Perform cross-multiplication:
- Multiply 21 by 50.
- Multiply 30 by 35.
2. Calculate the products:
- [tex]\(21 \times 50 = 1050\)[/tex]
- [tex]\(30 \times 35 = 1050\)[/tex]
3. Compare the products: [tex]\(1050 = 1050\)[/tex]
Since the products are equal, [tex]\(\frac{21}{30}\)[/tex] and [tex]\(\frac{35}{50}\)[/tex] do form a proportion.
Result: True
---
d. Check if [tex]\(\frac{1.8}{7.5}\)[/tex] and [tex]\(\frac{1.2}{5}\)[/tex] form a proportion.
1. Perform cross-multiplication:
- Multiply 1.8 by 5.
- Multiply 7.5 by 1.2.
2. Calculate the products:
- [tex]\(1.8 \times 5 = 9.0\)[/tex]
- [tex]\(7.5 \times 1.2 = 9.0\)[/tex]
3. Compare the products: [tex]\(9.0 = 9.0\)[/tex]
Since the products are equal, [tex]\(\frac{1.8}{7.5}\)[/tex] and [tex]\(\frac{1.2}{5}\)[/tex] do form a proportion.
Result: True
I hope this helps explain whether each pair of ratios forms a proportion!
