Given:
A, B, and C denote the mean, median, and mode of some data.
A : C = 10 : 7

To find:
The ratio B : A

Solution:

Mean (A):
The mean, denoted by A, is the average of all the data points. It is calculated by summing up all the values and dividing it by the total number of data points.

Median (B):
The median, denoted by B, is the middle value of a set of data points when they are arranged in ascending or descending order. If the number of data points is even, the median is the average of the two middle values.

Mode (C):
The mode, denoted by C, is the value that appears most frequently in a set of data points.

Since A : C = 10 : 7, we can write A = (10/7)C.

Key Points:
- The mean, median, and mode are measures of central tendency used to describe the average or typical value in a set of data.
- The mean is influenced by extreme values, while the median is not.
- The mode represents the value that occurs most frequently in the dataset.
- The mean and mode can have multiple values, but the median is unique.

To find the ratio B : A, we need to express B in terms of A.

Ratio B : A:
We know that the mean (A) is influenced by extreme values, while the median (B) is not. Therefore, the median is a more reliable measure of central tendency when there are extreme values or outliers in the dataset.

Let's consider an example to understand this better. Suppose we have a dataset: 1, 2, 3, 4, 100. The mean of this dataset is (1+2+3+4+100)/5 = 22, while the median is 3. Here, the extreme value 100 significantly affects the mean, making it larger than the typical value in the dataset. However, the median remains unaffected by extreme values.

Therefore, the ratio B : A is less than 1, indicating that the median is usually smaller than the mean.

So, the correct answer is 9 : 10.