Solving Number Problem: Finding the Smaller Number Given the Difference and Division
Understanding how to solve problems involving the difference between two numbers and their relationship through division is a critical skill in mathematics. These types of problems are common in various competitive exams and standardized tests. This article will walk you through several examples, illustrating step-by-step solutions to these types of problems.
Problem Statement 1
The difference between two numbers is 2730. On dividing the larger number by the smaller, we get a quotient of 12 and a remainder of 30. What is the smaller number?
Step-by-Step Solution
Step 1: Set Up the Problem
Let the smaller number be x. The larger number is then x 2730 (since the difference between the two numbers is 2730).
Step 2: Use the Division Property
When the larger number is divided by the smaller number, we get a quotient of 12 and a remainder of 30.
The equation can be written as:
Larger number Quotient × Smaller number Remainder
Or
x 2730 12x 30
Step 3: Solve the Equation
Rearrange the equation to isolate x:
x 2730 12x 30
2730 - 30 12x - x
2700 11x
x 2700 / 11 270
The smaller number is 270.
Problem Statement 2
Let the numbers be m and m - 1550.
Given the same conditions: the difference between the numbers is 2730, and when the larger number is divided by the smaller, the quotient is 12 and the remainder is 30.
Step-by-Step Solution
Step 1: Set Up the Problem
Let the smaller number be m - 1550. The larger number is then m.
Step 2: Use the Division Property
The division property gives us:
m 12(m - 1550) 30
Step 3: Solve the Equation
Rearrange the equation to isolate m:
m 12m - 18600 30
18630 11m
m 18630 / 11 1693.6363...
Therefore, the smaller number is 1693.6363... - 1550 143.6363....
Problem Statement 3
The difference between two numbers is 1097. The larger number divided by the smaller number gives a quotient of 10 and a remainder of 17. What is the smaller number?
Step-by-Step Solution
Step 1: Set Up the Problem
Let the smaller number be x. Then the larger number is 1 17.
Step 2: Use the Division Property
The division property gives us:
1 17 - x 1097
Step 3: Solve the Equation
Rearrange the equation to isolate x:
9x 17 1097
9x 1097 - 17
9x 1080
x 1080 / 9 120
The smaller number is 120.
Problem Statement 4
The difference between two numbers is 2730. On dividing the larger number by the smaller, we get 12 as the quotient and 30 as the remainder. What is the smaller number? (Alternative Approach)
Step-by-Step Solution
Step 1: Set Up the Problem
Let the smaller number be b. Then the larger number is 6b 15.
Step 2: Use the Difference Condition
The difference between the numbers is 2730:
6b 15 - b 2730
Step 3: Solve the Equation
Rearrange the equation to isolate b:
5b 15 2730
5b 2730 - 15
5b 2715
b 2715 / 5 543
The smaller number is b 543.
Problem Statement 5
The difference between two numbers is 1465. When the larger number is divided by the smaller, the quotient is 6 and the remainder is 15. What is the smaller number?
Step-by-Step Solution
Step 1: Set Up the Problem
Let the smaller number be x. Then the larger number is 6x 15.
Step 2: Use the Difference Condition
The difference between the numbers is 1465:
6x 15 - x 1465
Step 3: Solve the Equation
Rearrange the equation to isolate x:
5x 15 1465
5x 1465 - 15
5x 1450
x 1450 / 5 290
The smaller number is 290.
Conclusion
Solving number problems involving the difference and division is mainly about setting up equations based on the given conditions and solving them step-by-step. By understanding the relationship between the numbers and using the basic principles of arithmetic, you can easily find the required numbers.
For anyone preparing for competitive exams or interested in improving their mathematical skills, these types of problems are excellent practice. Keep practicing and you'll become proficient in solving such problems with ease.