Understanding Ratio Division for Rope Length

The question asks us to divide a rope of a specific length into two parts based on a given ratio and then find the length of the longer piece. The total length of the rope is 84 m, and it is divided in the ratio 7 : 5.

When a quantity is divided in a ratio, say $a : b$, it means the quantity is split into $a + b$ equal parts. One part gets $a$ of these parts, and the other part gets $b$ of these parts.

Calculating Total Parts in the Ratio

The given ratio is 7 : 5. To find the total number of equal parts the rope is divided into, we sum the numbers in the ratio:

Total parts $= 7 + 5 = 12$ parts.

Determining the Length of Each Part

The total length of the rope is 84 m. This total length corresponds to the 12 equal parts. To find the length of one part, we divide the total length by the total number of parts:

Length of one part $= \frac{\text{Total length}}{\text{Total parts}} = \frac{84 \text{ m}}{12 \text{ parts}}$

Length of one part $= 7 \text{ m/part}$.

Finding the Length of the Longer Piece

The ratio is 7 : 5. This means one piece corresponds to 7 parts and the other piece corresponds to 5 parts. The longer piece is the one with the larger number of parts, which is 7 parts.

To find the length of the longer piece, we multiply the length of one part by the number of parts in the longer piece:

Length of the longer piece $= \text{Length of one part} \times \text{Number of parts in longer piece}$

Length of the longer piece $= 7 \text{ m/part} \times 7 \text{ parts}$

Length of the longer piece $= 49 \text{ m}$.

We can also find the length of the shorter piece for completeness:

Length of the shorter piece $= \text{Length of one part} \times \text{Number of parts in shorter piece}$

Length of the shorter piece $= 7 \text{ m/part} \times 5 \text{ parts}$

Length of the shorter piece $= 35 \text{ m}$.

To verify, the sum of the lengths of the two pieces should equal the total length of the rope: $49 \text{ m} + 35 \text{ m} = 84 \text{ m}$. This confirms our calculation is correct.

Summary of Rope Division Calculation

The length of the longer piece is 49 m.

Revision Table: Ratio Division Basics

Here's a quick review of key terms related to ratio division:

• Ratio: A comparison of two or more quantities.
• Antecedent: The first term in a ratio (e.g., 7 in 7:5).
• Consequent: The second term in a ratio (e.g., 5 in 7:5).
• Ratio Sum: The sum of the terms in the ratio, representing the total number of parts.
• Unit Value: The value corresponding to one part of the ratio, calculated by dividing the total quantity by the ratio sum.

Additional Information: Applications of Ratios and Proportions

Ratios and proportions are fundamental concepts in mathematics with wide-ranging applications:

• Scaling: Used in maps, blueprints, and models to represent real-world sizes.
• Mixing: Used in recipes, chemistry, and construction to determine the amounts of different ingredients or components.
• Sharing: Used to divide quantities like money, property, or length among individuals or entities based on a defined ratio, as seen in this rope problem.
• Comparing Quantities: Ratios help compare quantities of the same unit or different units (rates).

Understanding how to work with ratios and proportions is crucial for solving many practical problems in various fields.