Problem 11 - Basic Engineering Mathematics

Question

Divide 126 in the ratio of 5 to 13 .

Step-by-Step Solution

Answer

The division results in 35 and 91.

Step 1: Understand the Problem

We need to divide the number 126 into two parts of a given ratio: 5:13. This means one part will be multiplied by 5 and the other by 13, so that together they sum up to 126.

Step 2: Calculate the Total Parts

Add together the parts of the ratio to determine the total number of parts the number 126 will be divided into. The total number of parts is: \(5 + 13 = 18\)

Step 3: Determine the Value of One Part

To find the value of one part, divide the total number, 126, by the total number of parts, 18. \(\text{Value of one part} = \frac{126}{18} = 7\)

Step 4: Calculate the First Share

Determine the amount corresponding to the first part of the ratio (5): \(\text{First Share} = 5 \times 7 = 35\)

Step 5: Calculate the Second Share

Determine the amount corresponding to the second part of the ratio (13): \(\text{Second Share} = 13 \times 7 = 91\)

Step 6: Verify the Solution

Add the two shares together to ensure they sum to the original number, 126.\(35 + 91 = 126\).The calculation is correct, as it matches the number to be divided.

Key Concepts

Division

To divide means to split a number into equal parts. It's one of the basic operations in mathematics. When we talk about division in the context of ratios, we break a whole into designated parts. For example, if you were asked to divide 126 into certain portions, division helps determine equal "units" or "shares" of a number. In the exercise, we divided 126, not into equal parts, but a specific ratio of 5:13. This means the larger part gets more and the smaller part gets less. To perform such a division, we figure out how many total parts are involved in the ratio and then how many equal units fit in each ratio part.

Partitive Proportion

Partitive proportion is a specific type of division problem where a number is divided into parts based on given ratios. The "parts" mean each one will be proportional, or "partitive," to others in the given ratio. Understandably, partitive proportion problems always start with understanding the given ratio first.

• First, calculate the sum of the ratio numbers to find the total number of parts. In our example, it's 5 + 13 = 18 parts.
• Next, divide the total quantity (126) by this sum to find how much each part is worth: 126 divided by 18 gives 7.
• Finally, multiply each ratio number by this unit to find each portion. Here, 5 parts and 13 parts multiplied by 7 yield 35 and 91 respectively.

This method helps distribute a total in exact proportions provided by the ratio.

Verification of Solutions

Verification is crucial in solving ratios and proportions to ensure every step is correct and faithfully represents the problem. Checking answers in mathematics is like proofreading in writing. Once you've split your number into calculated portions, you must add them back together.

• In the exercise, we divided into parts of 35 and 91, adding these gives back 126.
• Re-check the initial distribution to confirm it adheres to the original ratio, ensuring accuracy.

Verification reassures that the solution is precise. It is an essential final step to prevent mistakes and is a reliable way to reinforce understanding.