Understanding and Solving the Ratio Problem

The problem asks us to find the ratio of sand to cement in a mixture, given the ratio of sand to gravel and the ratio of gravel to cement. We are provided with two ratios:

• Ratio of sand to gravel: 7 : 8
• Ratio of gravel to cement: 6 : 7

To find the ratio of sand to cement, we need to connect these two ratios using the common element, which is gravel. The key is to make the amount representing gravel in both ratios the same.

Combining Ratios Step-by-Step

Let the ratio of sand to gravel be \(S:G\) and the ratio of gravel to cement be \(G:C\).

• Given: \(S:G = 7:8\)
• Given: \(G:C = 6:7\)

We want to find \(S:C\).

To combine these ratios, we need to find a common value for the gravel component in both ratios. The current values for gravel are 8 in the first ratio and 6 in the second ratio. We find the Least Common Multiple (LCM) of 8 and 6.

The multiples of 8 are 8, 16, 24, 32, ...

The multiples of 6 are 6, 12, 18, 24, 30, ...

The LCM of 8 and 6 is 24.

Now, we adjust each ratio so that the gravel part becomes 24.

For the ratio \(S:G = 7:8\), to make the gravel part 24, we need to multiply 8 by 3 (\(8 \times 3 = 24\)). So, we multiply both parts of this ratio by 3:

\(S:G = (7 \times 3) : (8 \times 3) = 21 : 24\)

For the ratio \(G:C = 6:7\), to make the gravel part 24, we need to multiply 6 by 4 (\(6 \times 4 = 24\)). So, we multiply both parts of this ratio by 4:

\(G:C = (6 \times 4) : (7 \times 4) = 24 : 28\)

Now we have the ratios where the gravel part is the same:

• \(S:G = 21:24\)
• \(G:C = 24:28\)

Since the gravel component is consistently represented by 24 in both new ratios, we can combine them to find the ratio of sand to gravel to cement \(S:G:C\).

\(S:G:C = 21:24:28\)

We are asked to find the ratio of sand to cement, which is \(S:C\). From the combined ratio \(21:24:28\), we can directly see the ratio of sand to cement.

\(S:C = 21:28\)

Finally, we simplify the ratio \(21:28\) by dividing both numbers by their greatest common divisor (GCD). The GCD of 21 and 28 is 7.

\(21 \div 7 = 3\)

\(28 \div 7 = 4\)

So, the simplified ratio of sand to cement is \(3:4\).

Summary of the Solution

Given:

• Sand : Gravel = 7 : 8
• Gravel : Cement = 6 : 7

Find LCM of the gravel parts (8 and 6), which is 24.

Adjust ratios:

• Sand : Gravel = \(7 \times 3 : 8 \times 3 = 21 : 24\)
• Gravel : Cement = \(6 \times 4 : 7 \times 4 = 24 : 28\)

Combined ratio Sand : Gravel : Cement = 21 : 24 : 28

Ratio of Sand : Cement = 21 : 28

Simplify the ratio: \(21 \div 7 : 28 \div 7 = 3 : 4\)

The final ratio of sand to cement is 3 : 4.

Revision Table - Key Ratio Concepts

Additional Information - Ratio and Proportion

Ratios are fundamental in mathematics and are used to compare quantities. A proportion is an equation stating that two ratios are equal. Understanding how to manipulate and combine ratios is crucial for solving problems involving mixtures, proportions, and scaling.

When combining ratios like \(A:B\) and \(B:C\), the common element is B. The process involves making the representation of B consistent across both ratios. If \(A:B = a:b\) and \(B:C = c:d\), we find the LCM of \(b\) and \(c\). Let the LCM be \(L\). We then multiply the first ratio by \(L/b\) and the second ratio by \(L/c\).

• \(A:B = a \times (L/b) : b \times (L/b) = a(L/b) : L\)
• \(B:C = c \times (L/c) : d \times (L/c) = L : d(L/c)\)

Now the combined ratio is \(A:B:C = a(L/b) : L : d(L/c)\). From this, \(A:C = a(L/b) : d(L/c)\), which can then be simplified.

In our specific problem:

• \(A=S\), \(B=G\), \(C=C\)
• \(a=7\), \(b=8\), \(c=6\), \(d=7\)
• LCM(8, 6) = 24. Here, \(L=24\).
• For S:G, \(L/b = 24/8 = 3\). New S:G = \(7 \times 3 : 8 \times 3 = 21:24\).
• For G:C, \(L/c = 24/6 = 4\). New G:C = \(6 \times 4 : 7 \times 4 = 24:28\).
• Combined S:G:C = 21:24:28.
• S:C = 21:28, simplified to 3:4.

This method ensures the relative proportions between all components are maintained correctly when combining the ratios.