If a cycle has front wheel of perimeter 65 and back wheel of 52. If front wheel revolves 456 times. How many revolutions will the back wheel take?

The back wheel will take 570 revolutions approximately.

Explanation

To solve this problem, we can use the concept that the distance covered by one revolution of each wheel is equal.

The distance covered by one revolution of the front wheel is equal to its perimeter, which is 65 units.

So, if the front wheel revolves 456 times, the total distance covered by the front wheel is:

456 * 65 = 29640

Now, since the distance covered by one revolution of the back wheel is also equal to its perimeter, which is 52 units, we can find the number of revolutions the back wheel will take by dividing the total distance covered by the back wheel's perimeter:

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Number of revolutions of back wheel = Total distance covered / Perimeter of back wheel Number of revolutions of back wheel = 29640 / 52 ≈ 570

So, the back wheel will take approximately 570 revolutions.

Proportional Reasoning in Mathematics

Proportional reasoning is a fundamental concept in mathematics that involves understanding and working with relationships between quantities that change in direct proportion to each other. In other words, when one quantity changes, the other quantity changes in a consistent manner.

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Here are some key aspects and examples of proportional reasoning in mathematics:

Definition of Proportionality: Two quantities "a" and "b" are said to be directly proportional if an increase in one quantity results in a proportional increase in the other quantity, and vice versa. Mathematically, this can be represented as "a ∝ b", or "a = kb", where "k" is the constant of proportionality.
Constant of Proportionality: The constant "k" represents the factor by which one quantity changes relative to the change in the other quantity. It remains the same throughout the relationship between the two quantities.
Proportional Relationships: Proportional relationships can be represented in various forms, such as equations, tables, graphs, and verbal descriptions.
- Equations: y = kx
- Tables: Organizing data in a table where one column represents the input variable (e.g., "x") and the other column represents the output variable (e.g., "y"). The values in the output column are obtained by multiplying the corresponding values in the input column by the constant of proportionality.
- Graphs: In a proportional relationship, the graph of "y" versus "x" is a straight line passing through the origin (0,0). The slope of this line represents the constant of proportionality.
- Verbal Descriptions: Statements indicating that one quantity is directly proportional to another quantity.
Applications:
- Geometry: Proportional reasoning is often used in geometry to solve problems involving similar figures. For example, if two triangles are similar, then their corresponding sides are proportional.
- Finance: Proportional reasoning is applied in finance when calculating interest, taxes, and other financial transactions. For instance, the amount of interest earned on an investment is directly proportional to the principal amount invested and the interest rate.
- Physics: Many physical phenomena involve proportional relationships. For instance, according to Hooke's Law, the force exerted by a spring is directly proportional to the amount it is stretched or compressed.