Calculating Sand to Cement Ratio from Mixture Proportions

This problem involves combining two different ratios that share a common component. We are given the ratio of sand to gravel and the ratio of gravel to cement. To find the ratio of sand to cement, we need to link these two ratios using the common component, which is gravel.

Understanding the Given Ratios

We are provided with two ratios:

• Ratio of Sand to Gravel is $\text{Sand} : \text{Gravel} = 3 : 4$. This can also be written as $\frac{\text{Sand}}{\text{Gravel}} = \frac{3}{4}$.
• Ratio of Gravel to Cement is $\text{Gravel} : \text{Cement} = 6 : 7$. This can also be written as $\frac{\text{Gravel}}{\text{Cement}} = \frac{6}{7}$.

Combining Ratios using the Common Term (Gravel)

To find the ratio of Sand to Cement, we need to make the 'Gravel' part of both ratios consistent. The current gravel proportions are 4 and 6. We find the least common multiple (LCM) of 4 and 6.

The multiples of 4 are 4, 8, 12, 16, ...

The multiples of 6 are 6, 12, 18, ...

The LCM of 4 and 6 is 12.

Adjusting the Ratios

Now, we adjust each original ratio so that the 'Gravel' part becomes 12.

1. Adjusting Sand : Gravel (3 : 4): To change the gravel part from 4 to 12, we multiply by $\frac{12}{4} = 3$. We must multiply both parts of the ratio by 3 to maintain the proportion: $$\text{Sand} : \text{Gravel} = (3 \times 3) : (4 \times 3) = 9 : 12$$
2. Adjusting Gravel : Cement (6 : 7): To change the gravel part from 6 to 12, we multiply by $\frac{12}{6} = 2$. We must multiply both parts of the ratio by 2 to maintain the proportion: $$\text{Gravel} : \text{Cement} = (6 \times 2) : (7 \times 2) = 12 : 14$$

Determining the Ratio of Sand to Cement

Now that the 'Gravel' part in both adjusted ratios is 12, we can see the combined ratio of Sand : Gravel : Cement is 9 : 12 : 14.

From this combined ratio, the ratio of Sand to Cement is simply the Sand part compared to the Cement part.

Therefore, the ratio of sand to cement in the mixture is 9 : 14.

Revision Table: Ratio Concepts

Additional Information: Applications of Ratios

Ratios are fundamental in mathematics and have many real-world applications, including:

• Mixtures and Solutions: Determining the proportion of different substances in a compound or solution, as seen in this sand, gravel, and cement example.
• Scaling and Maps: Representing distances on maps or scaling blueprints where a small measurement represents a larger real-world distance.
• Cooking and Recipes: Adjusting ingredient quantities proportionally when scaling recipes up or down.
• Finance: Analyzing financial statements using ratios like debt-to-equity ratio or price-to-earnings ratio.
• Physics and Engineering: Describing relationships between physical quantities, such as speed (distance to time) or density (mass to volume).

Understanding how to work with ratios, especially combining them as demonstrated in this problem, is crucial for solving various quantitative problems.