Answer :
Sure! Let's solve each part of the problem step by step.
### Part 1: Cost of the Book
1. Initial Cost: The initial cost of the book is 100 lei.
2. Reduction: There is a 10% reduction in the book's price. To find the reduced price, calculate 10% of 100:
[tex]\[
\text{Reduction amount} = 0.10 \times 100 = 10 \text{ lei}
\][/tex]
Subtract this reduction from the original price:
[tex]\[
\text{Reduced price} = 100 - 10 = 90 \text{ lei}
\][/tex]
3. Increase: After the reduction, the price increases by 10% of the new price (90 lei). Calculate 10% of 90:
[tex]\[
\text{Increase amount} = 0.10 \times 90 = 9 \text{ lei}
\][/tex]
Add this increase to the reduced price:
[tex]\[
\text{Final price} = 90 + 9 = 99 \text{ lei}
\][/tex]
Therefore, the final price of the book, after the reduction and increase, is 99 lei.
### Part 2: Number of Natural Numbers Divisible by 8 with Quotient 6
1. Understanding the Problem:
We need to find numbers that, when divided by 8, give a quotient of 6. The equation for such a number [tex]\( n \)[/tex] can be written as:
[tex]\[
n = 8 \times 6 + r = 48 + r
\][/tex]
where [tex]\( r \)[/tex] is the remainder when [tex]\( n \)[/tex] is divided by 8.
2. Remainder Range:
The remainder [tex]\( r \)[/tex] can be from 0 to 7 (since these are all possible remainders when dividing by 8).
3. Counting Possible Numbers:
The numbers are: 48 (when [tex]\( r=0 \)[/tex]), 49 (when [tex]\( r=1 \)[/tex]), 50 (when [tex]\( r=2 \)[/tex]), 51 (when [tex]\( r=3 \)[/tex]), 52 (when [tex]\( r=4 \)[/tex]), 53 (when [tex]\( r=5 \)[/tex]), 54 (when [tex]\( r=6 \)[/tex]), and 55 (when [tex]\( r=7 \)[/tex]).
So, there are 8 natural numbers where the division by 8 gives a quotient of 6.
In summary:
- The final price of the book is 99 lei.
- The number of natural numbers which give a quotient of 6 when divided by 8 is 8.
### Part 1: Cost of the Book
1. Initial Cost: The initial cost of the book is 100 lei.
2. Reduction: There is a 10% reduction in the book's price. To find the reduced price, calculate 10% of 100:
[tex]\[
\text{Reduction amount} = 0.10 \times 100 = 10 \text{ lei}
\][/tex]
Subtract this reduction from the original price:
[tex]\[
\text{Reduced price} = 100 - 10 = 90 \text{ lei}
\][/tex]
3. Increase: After the reduction, the price increases by 10% of the new price (90 lei). Calculate 10% of 90:
[tex]\[
\text{Increase amount} = 0.10 \times 90 = 9 \text{ lei}
\][/tex]
Add this increase to the reduced price:
[tex]\[
\text{Final price} = 90 + 9 = 99 \text{ lei}
\][/tex]
Therefore, the final price of the book, after the reduction and increase, is 99 lei.
### Part 2: Number of Natural Numbers Divisible by 8 with Quotient 6
1. Understanding the Problem:
We need to find numbers that, when divided by 8, give a quotient of 6. The equation for such a number [tex]\( n \)[/tex] can be written as:
[tex]\[
n = 8 \times 6 + r = 48 + r
\][/tex]
where [tex]\( r \)[/tex] is the remainder when [tex]\( n \)[/tex] is divided by 8.
2. Remainder Range:
The remainder [tex]\( r \)[/tex] can be from 0 to 7 (since these are all possible remainders when dividing by 8).
3. Counting Possible Numbers:
The numbers are: 48 (when [tex]\( r=0 \)[/tex]), 49 (when [tex]\( r=1 \)[/tex]), 50 (when [tex]\( r=2 \)[/tex]), 51 (when [tex]\( r=3 \)[/tex]), 52 (when [tex]\( r=4 \)[/tex]), 53 (when [tex]\( r=5 \)[/tex]), 54 (when [tex]\( r=6 \)[/tex]), and 55 (when [tex]\( r=7 \)[/tex]).
So, there are 8 natural numbers where the division by 8 gives a quotient of 6.
In summary:
- The final price of the book is 99 lei.
- The number of natural numbers which give a quotient of 6 when divided by 8 is 8.
