Solving Ratio Problems: Finding an Unknown Value

This problem involves ratios and finding an unknown value when one value in a ratio is given. We are given two ratios, $\small u:v$ and $\small v:w$, and the value of $\small u$. We need to find the value of $\small w$. To solve this, we will use the given information step-by-step.

Understanding the Given Ratios and Value

We are provided with the following information:

• Ratio 1: $\small u : v = 4 : 7$
• Ratio 2: $\small v : w = 9 : 7$
• Value of $\small u = 72$

Our goal is to find the value of $\small w$. Notice that the variable $\small v$ is common to both ratios. This is key to connecting $\small u$ and $\small w$.

Step 1: Finding the Value of v

We know the ratio $\small u : v$ is $\small 4 : 7$. This can be written as a fraction:

$\small \frac{u}{v} = \frac{4}{7}$

We are given that $\small u = 72$. We can substitute this value into the equation:

$\small \frac{72}{v} = \frac{4}{7}$

To find $\small v$, we can cross-multiply:

$\small 4 \times v = 72 \times 7$

$\small 4v = 504$

Now, divide both sides by 4 to isolate $\small v$:

$\small v = \frac{504}{4}$

$\small v = 126$

So, the value of $\small v$ is 126.

Step 2: Finding the Value of w

Now that we have the value of $\small v$, we can use the second ratio $\small v : w = 9 : 7$ to find $\small w$. This ratio can also be written as a fraction:

$\small \frac{v}{w} = \frac{9}{7}$

Substitute the value of $\small v = 126$ that we just found:

$\small \frac{126}{w} = \frac{9}{7}$

Again, cross-multiply to solve for $\small w$:

$\small 9 \times w = 126 \times 7$

$\small 9w = 882$

Now, divide both sides by 9 to isolate $\small w$:

$\small w = \frac{882}{9}$

$\small w = 98$

Thus, the value of $\small w$ is 98.

Summary of Values Found

Let's list the values we determined:

• $\small u = 72$ (given)
• $\small v = 126$ (calculated)
• $\small w = 98$ (calculated)

Verification using Ratios

Let's check if these values satisfy the original ratios:

• Is $\small u : v = 72 : 126$ equivalent to $\small 4 : 7$?

  Divide both 72 and 126 by their greatest common divisor, which is 18. $\small 72 \div 18 = 4$ and $\small 126 \div 18 = 7$. So, $\small 72 : 126$ simplifies to $\small 4 : 7$. This matches the first ratio.

• Is $\small v : w = 126 : 98$ equivalent to $\small 9 : 7$?

  Divide both 126 and 98 by their greatest common divisor, which is 14. $\small 126 \div 14 = 9$ and $\small 98 \div 14 = 7$. So, $\small 126 : 98$ simplifies to $\small 9 : 7$. This matches the second ratio.

The values satisfy both original ratios.

Based on our calculations, when $\small u = 72$, the value of $\small w$ is 98.

Ratio Problems: Revision Table

Additional Information on Ratio Calculations

When dealing with multiple ratios involving common terms, there are two main ways to approach the problem:

1. Sequential Calculation: This is the method we used. Use the first ratio and the known value to find the value of the common term ($\small v$). Then, use the second ratio and the calculated value of the common term ($\small v$) to find the value of the final unknown ($\small w$). This method is often straightforward when you have a clear chain of dependencies.
2. Combining Ratios: This involves finding a common representation for the shared term (like $\small v$ in this case) across both ratios to establish a single combined ratio (like $\small u:v:w$). This is done by finding a common multiple for the parts representing the shared term in each ratio and scaling the entire ratio accordingly. Once the combined ratio is found, you can determine the value of one 'part' based on the given value of one variable and then calculate the values of the other variables. While not used in the primary step-by-step solution above, the Revision Table shows how combining ratios $\small u:v=4:7$ and $\small v:w=9:7$ gives $\small u:v:w = 36:63:49$. If $\small u=72$ (which is 36 parts), then each part is $\small 72/36=2$. Thus $\small w$ (49 parts) is $\small 49 \times 2 = 98$. Both methods yield the same result and are valid approaches to solving ratio problems.