Answer :
Sure! Let's go through each question step by step:
a. Multiply [tex]\(1.37 \times 10^2\)[/tex]:
To solve this, you multiply 1.37 by 100. This is because [tex]\(10^2\)[/tex] means 10 multiplied by itself, which is 100.
[tex]\[
1.37 \times 100 = 137
\][/tex]
So, the answer is 137.
b. If 10 kg of apples cost Rs. 3500, what is the cost of 12 kg of apples at the same rate?
First, find the cost per kilogram. To do this, divide the total cost of 10 kg by 10.
[tex]\[
\text{Cost per kg} = \frac{3500}{10} = 350
\][/tex]
Next, multiply the cost per kg by 12 to find the cost for 12 kg.
[tex]\[
\text{Cost for 12 kg} = 350 \times 12 = 4200
\][/tex]
So, the cost of 12 kg of apples is Rs. 4200.
c. Convert the repeating decimal [tex]\(1.\overline{57}\)[/tex] to a fraction:
The decimal [tex]\(1.\overline{57}\)[/tex] means that 57 is repeating. Let's convert it to a fraction.
Let [tex]\( x = 1.575757...\)[/tex]
Multiply both sides by 100 to move the repeating part:
[tex]\[
100x = 157.575757...
\][/tex]
Subtract the original [tex]\( x = 1.575757...\)[/tex] from this equation:
[tex]\[
100x - x = 157.575757... - 1.575757...
\][/tex]
This simplifies to:
[tex]\[
99x = 156
\][/tex]
Now, solve for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{156}{99}
\][/tex]
To simplify, divide both numerator and denominator by their greatest common divisor, which is 3:
[tex]\[
x = \frac{52}{33}
\][/tex]
So, the fraction is [tex]\(\frac{52}{33}\)[/tex].
d. Convert the decimal number 137 to base 5:
To convert 137 from decimal to base 5, we divide by 5 repeatedly and keep track of the remainders.
1. [tex]\(137 \div 5 = 27\)[/tex] remainder 2
2. [tex]\(27 \div 5 = 5\)[/tex] remainder 2
3. [tex]\(5 \div 5 = 1\)[/tex] remainder 0
4. [tex]\(1 \div 5 = 0\)[/tex] remainder 1
Now, read the remainders from the last to the first to get the number in base 5:
So, 137 in base 5 is [tex]\(1022_5\)[/tex].
If you have any more questions, feel free to ask!
a. Multiply [tex]\(1.37 \times 10^2\)[/tex]:
To solve this, you multiply 1.37 by 100. This is because [tex]\(10^2\)[/tex] means 10 multiplied by itself, which is 100.
[tex]\[
1.37 \times 100 = 137
\][/tex]
So, the answer is 137.
b. If 10 kg of apples cost Rs. 3500, what is the cost of 12 kg of apples at the same rate?
First, find the cost per kilogram. To do this, divide the total cost of 10 kg by 10.
[tex]\[
\text{Cost per kg} = \frac{3500}{10} = 350
\][/tex]
Next, multiply the cost per kg by 12 to find the cost for 12 kg.
[tex]\[
\text{Cost for 12 kg} = 350 \times 12 = 4200
\][/tex]
So, the cost of 12 kg of apples is Rs. 4200.
c. Convert the repeating decimal [tex]\(1.\overline{57}\)[/tex] to a fraction:
The decimal [tex]\(1.\overline{57}\)[/tex] means that 57 is repeating. Let's convert it to a fraction.
Let [tex]\( x = 1.575757...\)[/tex]
Multiply both sides by 100 to move the repeating part:
[tex]\[
100x = 157.575757...
\][/tex]
Subtract the original [tex]\( x = 1.575757...\)[/tex] from this equation:
[tex]\[
100x - x = 157.575757... - 1.575757...
\][/tex]
This simplifies to:
[tex]\[
99x = 156
\][/tex]
Now, solve for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{156}{99}
\][/tex]
To simplify, divide both numerator and denominator by their greatest common divisor, which is 3:
[tex]\[
x = \frac{52}{33}
\][/tex]
So, the fraction is [tex]\(\frac{52}{33}\)[/tex].
d. Convert the decimal number 137 to base 5:
To convert 137 from decimal to base 5, we divide by 5 repeatedly and keep track of the remainders.
1. [tex]\(137 \div 5 = 27\)[/tex] remainder 2
2. [tex]\(27 \div 5 = 5\)[/tex] remainder 2
3. [tex]\(5 \div 5 = 1\)[/tex] remainder 0
4. [tex]\(1 \div 5 = 0\)[/tex] remainder 1
Now, read the remainders from the last to the first to get the number in base 5:
So, 137 in base 5 is [tex]\(1022_5\)[/tex].
If you have any more questions, feel free to ask!
