Question

Divide 1 in the ratio \(6: 3\),

Short answer

Question: Divide 1 in the ratio 6:3, and give the value of each part. Answer: When divided in the ratio 6:3, the first part will be 2/3, and the second part will be 1/3.

Step-by-step solution

Simplify the ratio

First, let's simplify the given ratio \(6:3\). Divide both parts of the ratio by their greatest common divisor (3) to get the simplified ratio as follows: \(6:3 = \frac{6}{3} : \frac{3}{3} = 2:1\)

Calculate the total share

Now, calculate the total share of the ratio by adding both parts of the simplified ratio: Total share = \(2 + 1 = 3\)

Divide 1 according to the shares

Next, we will find the part of 1 corresponding to each part of the simplified ratio. To do this, we multiply the share of each part by the total value (1) and divide by the total share (3): First part: \(\frac{2}{3} × 1 = \frac{2}{3}\) Second part: \(\frac{1}{3} × 1 = \frac{1}{3}\)

Final Answer

The value of 1 divided into the simplified ratio is: First part: \(\frac{2}{3}\) Second part: \(\frac{1}{3}\)

Key concepts

Simplifying Ratios

When dealing with ratios, simplifying them makes calculations easier and the results more understandable. Simplifying a ratio involves expressing it in the smallest whole numbers that can represent the same relationship. This is akin to reducing fractions to their simplest form.

To illustrate, consider the exercise where we have the ratio of two quantities as '6:3'. This ratio indicates that for every 6 units of the first quantity, there are 3 units of the second quantity. However, this ratio can be simplified by determining the largest number that divides both quantities without leaving a remainder, known as the greatest common divisor (GCD).

In this case, both 6 and 3 can be divided evenly by 3, making it the GCD. By dividing each part of the ratio by 3, we obtain the simplified ratio of '2:1'. This simplified form conveys the same proportional relationship in a clearer and more concise manner—now it tells us that for every 2 units of the first quantity, there is 1 unit of the second quantity.

Greatest Common Divisor

The greatest common divisor (GCD), also known as the greatest common factor (GCF), plays an essential role in simplifying ratios. It is the largest positive integer that divides the numbers without leaving a remainder.

For simplification purposes, the GCD helps reduce ratios to their simplest form. This makes it easier to work with those ratios, whether comparing different quantities or dividing a sum proportionally. In the provided exercise, the GCD of 6 and 3 is 3. By dividing both 6 and 3 by their GCD of 3, we simplified the ratio effectively.

To find the GCD, various methods can be employed, such as listing factors, using the Euclidean algorithm, or prime factorization. Once the GCD is known, we divide both parts of the ratio by the GCD to obtain the simplified ratio. Remembering that the GCD must be a positive integer, and if the GCD is 1, the ratio is already in its simplest form.

Proportional Division

Proportional division is a method of dividing quantities so that the parts maintain a specific ratio or relationship to each other.

In the exercise example, the goal was to divide the number 1 proportionally according to the ratio '6:3', which simplifies to '2:1'. Proportional division ensures that 1 is divided into parts that are in proportion to each other as per the simplified ratio.

First, we calculate the sum of the parts of the ratio (also known as the 'total share')—2+1 to get 3. This sum acts as the denominator when we work out the proportional parts. Then, the value to be divided (in this case, 1) is multiplied by each share of the ratio and divided by the total share: \(\frac{2}{3} \times 1 = \frac{2}{3}\) for the first part, and \(\frac{1}{3} \times 1 = \frac{1}{3}\) for the second part.

This division method is particularly valuable when dealing with problems related to mixtures, profit sharing, or any situation requiring a division in a particular proportion, ensuring fairness and balance in the distribution.