Answer :
To solve the problem of multiplying [tex]\(9 \, \text{kg} \, 3 \, \text{hg} \, 1 \, \text{cg}\)[/tex] by 9, let's break it down into simpler steps.
1. Understand the units:
- Kilograms (kg), hectograms (hg), and centigrams (cg) are units of mass.
- 1 kg = 1000 grams (g)
- 1 hg = 100 grams (g)
- 1 cg = 0.01 grams (g)
2. Convert each unit to decigrams (dg):
- Since 1 gram = 10 decigrams, we will convert everything into decigrams for easier multiplication.
- Convert kilograms to decigrams:
[tex]\[
9 \, \text{kg} = 9 \times 1000 \times 10 = 90,000 \, \text{dg}
\][/tex]
- Convert hectograms to decigrams:
[tex]\[
3 \, \text{hg} = 3 \times 100 \times 10 = 3,000 \, \text{dg}
\][/tex]
- Convert centigrams to decigrams:
[tex]\[
1 \, \text{cg} = 1 \times 0.1 = 0.1 \, \text{dg}
\][/tex]
3. Calculate the total mass in decigrams:
[tex]\[
90,000 \, \text{dg} + 3,000 \, \text{dg} + 0.1 \, \text{dg} = 93,000.1 \, \text{dg}
\][/tex]
4. Multiply the total mass by 9:
[tex]\[
93,000.1 \times 9 = 837,000.9 \, \text{dg}
\][/tex]
5. Convert back to kg, hg, and cg:
- Convert decigrams to kilograms:
[tex]\[
\text{Kilograms} = \frac{837,000.9}{10,000} = 83.7 \, \text{kg}
\][/tex]
Extract the whole number as kilograms. So, 83 kg.
- Find remaining hectograms:
[tex]\[
\text{Remainder after kg} = 837,000.9 - (83 \times 10,000) = 7,000.9 \, \text{dg}
\][/tex]
[tex]\[
\text{Hectograms} = \frac{7,000.9}{1,000} = 7.0009 \, \text{hg}
\][/tex]
Again, extract the whole number, which is 7 hg.
- Find remaining centigrams:
[tex]\[
\text{Remainder after hg} = 7,000.9 - (7 \times 1,000) = 0.9 \, \text{dg}
\][/tex]
[tex]\[
\text{Centigrams} = 0.9 \times 10 = 9 \, \text{cg}
\][/tex]
6. Compile the final answer:
Therefore, the product of [tex]\(9 \, \text{kg} \, 3 \, \text{hg} \, 1 \, \text{cg}\)[/tex] by 9 is:
- 83 kg 7 hg 9 cg
So, the correct answer is [tex]\(83 \, \text{kg} \, 7 \, \text{hg} \, 9 \, \text{cg}\)[/tex].
1. Understand the units:
- Kilograms (kg), hectograms (hg), and centigrams (cg) are units of mass.
- 1 kg = 1000 grams (g)
- 1 hg = 100 grams (g)
- 1 cg = 0.01 grams (g)
2. Convert each unit to decigrams (dg):
- Since 1 gram = 10 decigrams, we will convert everything into decigrams for easier multiplication.
- Convert kilograms to decigrams:
[tex]\[
9 \, \text{kg} = 9 \times 1000 \times 10 = 90,000 \, \text{dg}
\][/tex]
- Convert hectograms to decigrams:
[tex]\[
3 \, \text{hg} = 3 \times 100 \times 10 = 3,000 \, \text{dg}
\][/tex]
- Convert centigrams to decigrams:
[tex]\[
1 \, \text{cg} = 1 \times 0.1 = 0.1 \, \text{dg}
\][/tex]
3. Calculate the total mass in decigrams:
[tex]\[
90,000 \, \text{dg} + 3,000 \, \text{dg} + 0.1 \, \text{dg} = 93,000.1 \, \text{dg}
\][/tex]
4. Multiply the total mass by 9:
[tex]\[
93,000.1 \times 9 = 837,000.9 \, \text{dg}
\][/tex]
5. Convert back to kg, hg, and cg:
- Convert decigrams to kilograms:
[tex]\[
\text{Kilograms} = \frac{837,000.9}{10,000} = 83.7 \, \text{kg}
\][/tex]
Extract the whole number as kilograms. So, 83 kg.
- Find remaining hectograms:
[tex]\[
\text{Remainder after kg} = 837,000.9 - (83 \times 10,000) = 7,000.9 \, \text{dg}
\][/tex]
[tex]\[
\text{Hectograms} = \frac{7,000.9}{1,000} = 7.0009 \, \text{hg}
\][/tex]
Again, extract the whole number, which is 7 hg.
- Find remaining centigrams:
[tex]\[
\text{Remainder after hg} = 7,000.9 - (7 \times 1,000) = 0.9 \, \text{dg}
\][/tex]
[tex]\[
\text{Centigrams} = 0.9 \times 10 = 9 \, \text{cg}
\][/tex]
6. Compile the final answer:
Therefore, the product of [tex]\(9 \, \text{kg} \, 3 \, \text{hg} \, 1 \, \text{cg}\)[/tex] by 9 is:
- 83 kg 7 hg 9 cg
So, the correct answer is [tex]\(83 \, \text{kg} \, 7 \, \text{hg} \, 9 \, \text{cg}\)[/tex].
