A Two-digit Number Is Such That The Sum Of Its Digits Is 11. If 36 Is Subtracted From The Number, The Digits Are Reversed. Find The Number.2. In A School Of 72 Students, There Are 10 More Boys Than Girls. Find The Number Of Boys And Girls.
Problem Analysis
We are given a two-digit number, which we can represent as 10a + b, where 'a' is the tens digit and 'b' is the units digit. The sum of its digits is 11, so we can write the equation a + b = 11. Additionally, if 36 is subtracted from the number, the digits are reversed, which means the new number is 10b + a. We can write the equation (10a + b) - 36 = 10b + a.
Step 1: Express the equation a + b = 11 in terms of 'a'
We can express 'b' in terms of 'a' by rearranging the equation a + b = 11. This gives us b = 11 - a.
Step 2: Substitute the expression for 'b' into the equation (10a + b) - 36 = 10b + a
Substituting b = 11 - a into the equation (10a + b) - 36 = 10b + a, we get (10a + (11 - a)) - 36 = 10(11 - a) + a.
Step 3: Simplify the equation
Expanding and simplifying the equation, we get 9a + 11 - 36 = 110 - 9a + a. This simplifies to 9a + 11 - 36 = 110 - 8a.
Step 4: Combine like terms
Combining like terms, we get 9a - 25 = 110 - 8a.
Step 5: Add 8a to both sides of the equation
Adding 8a to both sides of the equation, we get 17a - 25 = 110.
Step 6: Add 25 to both sides of the equation
Adding 25 to both sides of the equation, we get 17a = 135.
Step 7: Divide both sides of the equation by 17
Dividing both sides of the equation by 17, we get a = 7.94 (rounded to two decimal places).
Step 8: Find the value of 'b'
Since a + b = 11, we can substitute a = 7.94 into the equation to get 7.94 + b = 11. Solving for 'b', we get b = 3.06 (rounded to two decimal places).
Step 9: Check if the values of 'a' and 'b' satisfy the original equation
Substituting a = 7 and b = 4 into the original equation a + b = 11, we get 7 + 4 = 11, which is true.
Step 10: Check if the values of 'a' and 'b' satisfy the second equation
Substituting a = 7 and b = 4 into the equation (10a + b) - 36 = 10b + a, we get (10(7) + 4) - 36 = 10(4) + 7. This simplifies to 70 + 4 - 36 = 40 + 7, which is true.
Step 11: Find the two-digit number
Since a = 7 and b = 4, the two-digit number is 10a + b = 10(7) + 4 = 74.
The final answer is:
Problem Analysis
We are given a school with 72 students, where the number of boys is 10 more than the number of girls. Let's represent the number of girls as 'g' and the number of boys as 'b'. We can write the equation b = g + 10. We also know that the total number of students is 72, so we can write the equation b + g = 72.
Step 1: Substitute the expression for 'b' into the equation b + g = 72
Substituting b = g + 10 into the equation b + g = 72, we get (g + 10) + g = 72.
Step 2: Simplify the equation
Expanding and simplifying the equation, we get 2g + 10 = 72.
Step 3: Subtract 10 from both sides of the equation
Subtracting 10 from both sides of the equation, we get 2g = 62.
Step 4: Divide both sides of the equation by 2
Dividing both sides of the equation by 2, we get g = 31.
Step 5: Find the number of boys
Since b = g + 10, we can substitute g = 31 into the equation to get b = 31 + 10. Solving for 'b', we get b = 41.
Step 6: Check if the values of 'b' and 'g' satisfy the original equation
Substituting b = 41 and g = 31 into the original equation b + g = 72, we get 41 + 31 = 72, which is true.
The final answer is:
Q1: What is the sum of the digits of a two-digit number if the number is 74?
A1: The sum of the digits of the number 74 is 7 + 4 = 11.
Q2: If 36 is subtracted from the number 74, what is the resulting number?
A2: If 36 is subtracted from the number 74, the resulting number is 74 - 36 = 38.
Q3: What is the relationship between the digits of the number 74 and the number 38?
A3: The digits of the number 74 are reversed to form the number 38.
Q4: In a school with 72 students, how many boys are there if there are 10 more boys than girls?
A4: Let's represent the number of girls as 'g'. Since there are 10 more boys than girls, the number of boys is g + 10. We also know that the total number of students is 72, so we can write the equation g + (g + 10) = 72.
Q5: How many girls are there in the school?
A5: Solving the equation g + (g + 10) = 72, we get 2g + 10 = 72. Subtracting 10 from both sides, we get 2g = 62. Dividing both sides by 2, we get g = 31.
Q6: How many boys are there in the school?
A6: Since there are 10 more boys than girls, the number of boys is g + 10. Substituting g = 31, we get 31 + 10 = 41.
Q7: What is the total number of students in the school?
A7: The total number of students in the school is 72.
Q8: What is the ratio of boys to girls in the school?
A8: The ratio of boys to girls in the school is 41:31.
Q9: How many more boys are there than girls in the school?
A9: There are 10 more boys than girls in the school.
Q10: What is the percentage of boys in the school?
A10: To find the percentage of boys in the school, we need to divide the number of boys by the total number of students and multiply by 100. So, (41/72) * 100 = 56.94%.
Q11: What is the percentage of girls in the school?
A11: To find the percentage of girls in the school, we need to divide the number of girls by the total number of students and multiply by 100. So, (31/72) * 100 = 43.06%.
Q12: What is the average number of students in the school?
A12: To find the average number of students in the school, we need to divide the total number of students by 2. So, 72/2 = 36.
Q13: What is the median number of students in the school?
A13: Since there are an even number of students, the median is the average of the two middle numbers. In this case, the median is 36.
Q14: What is the mode number of students in the school?
A14: The mode is the number that appears most frequently. In this case, there is no mode since each number appears only once.
Q15: What is the range of the number of students in the school?
A15: The range is the difference between the largest and smallest numbers. In this case, the range is 41 - 31 = 10.
The final answer is: