Finding the Ratio A:B:C from the Equation 3A = 6B = 7C

This problem requires us to find the ratio of three variables A, B, and C, given an equation relating them. The key is to express each variable in terms of a common value and then find the simplest form of their ratio.

Steps to Solve for the Ratio A:B:C

We are given the equation:

\[3A = 6B = 7C\]

To find the ratio A : B : C, we can set each part of the equality equal to a constant value, say \(k\). This allows us to express A, B, and C individually in terms of this constant \(k\).

• From \(3A = k\), we get \(A = \frac{k}{3}\).
• From \(6B = k\), we get \(B = \frac{k}{6}\).
• From \(7C = k\), we get \(C = \frac{k}{7}\).

Now, we can write the ratio A : B : C as:

\[A : B : C = \frac{k}{3} : \frac{k}{6} : \frac{k}{7}\]

Since \(k\) is a common factor in all terms of the ratio (assuming \(k \neq 0\)), we can cancel \(k\) from each term:

\[A : B : C = \frac{1}{3} : \frac{1}{6} : \frac{1}{7}\]

To express this ratio in its simplest form, where the terms are integers, we need to multiply each fraction by the Least Common Multiple (LCM) of the denominators (3, 6, and 7).

Let's find the LCM of 3, 6, and 7:

• Prime factorization of 3 is 3.
• Prime factorization of 6 is \(2 \times 3\).
• Prime factorization of 7 is 7.

LCM(3, 6, 7) = \(2 \times 3 \times 7 = 42\).

Now, multiply each term in the ratio \(\frac{1}{3} : \frac{1}{6} : \frac{1}{7}\) by 42:

• First term: \(\frac{1}{3} \times 42 = 14\).
• Second term: \(\frac{1}{6} \times 42 = 7\).
• Third term: \(\frac{1}{7} \times 42 = 6\).

So, the ratio A : B : C is 14 : 7 : 6.

Verifying the Ratio

We can verify the ratio 14 : 7 : 6 using the original equation \(3A = 6B = 7C\). Let \(A=14x\), \(B=7x\), and \(C=6x\) for some common factor \(x\).

• \(3A = 3 \times (14x) = 42x\)
• \(6B = 6 \times (7x) = 42x\)
• \(7C = 7 \times (6x) = 42x\)

Since \(3A = 6B = 7C = 42x\), the ratio 14 : 7 : 6 is correct.

Final Ratio Result

The ratio A : B : C is 14 : 7 : 6.

Revision Table: Key Concepts for Ratio Problems

Additional Information: Understanding Ratios and Proportions

Ratios and proportions are fundamental concepts in mathematics used to compare quantities. A ratio expresses how much of one quantity there is compared to another. For example, a ratio of 2:3 for boys to girls means for every 2 boys, there are 3 girls.

When you have an equation like \(3A = 6B = 7C\), it implies a proportional relationship between the variables. If you increase A, B and C must also increase proportionally to maintain the equality.

The method of setting the expression equal to a constant \(k\) is a powerful technique for solving problems where multiple quantities are in a continued equality. It helps in isolating each variable and then forming the ratio easily. Remember that the ratio \(A:B:C\) represents the simplest form of the relationship between \(A\), \(B\), and \(C\). If \(A:B:C = p:q:r\), it means \(A = px\), \(B = qx\), and \(C = rx\) for some common factor \(x\).

Common mistakes include directly taking the coefficients (3:6:7) or their reciprocals without finding the LCM correctly. Always convert the ratio of fractions to integers by multiplying by the LCM of the denominators.