M=840; a:b=7:14

To solve a problem involving the ratio \( a:b = 7:14 \) and given \( M = 840 \), we first need to express \( a \) and \( b \) in terms of a common variable based on their ratio.

The ratio \( 7:14 \) can be simplified to \( 1:2 \). This means that for every 1 part of \( a \), there are 2 parts of \( b \). We can express \( a \) and \( b \) as:

\[ a = 1k \] \[ b = 2k \]

where \( k \) is a common multiplier.

Next, if we know that the sum of \( a \) and \( b \) equals \( M \), we can write:

\[ a + b = M \]

Substituting for \( a \) and \( b \):

\[ k + 2k = 840 \]

This simplifies to:

\[ 3k = 840 \]

Now, we can solve for \( k \):

\[ k = \frac{840}{3} = 280 \]

With \( k \) found, we can now calculate \( a \) and \( b \):

\[ a = 1k = 280 \] \[ b = 2k = 2 \times 280 = 560 \]

So, the values of \( a \) and \( b \) are:

\[ a = 280, \quad b = 560 \]