Certainly! Let's solve the problem step-by-step:
Given the ratio [tex]\( a: b = 4: 5 \)[/tex].
This means [tex]\( a = 4k \)[/tex] and [tex]\( b = 5k \)[/tex] for some constant [tex]\( k \)[/tex].
We need to find the ratio [tex]\( a+b : 2a - b \)[/tex].
First, let's calculate [tex]\( a + b \)[/tex]:
[tex]\[ a + b = 4k + 5k = 9k \][/tex]
Next, let's calculate [tex]\( 2a - b \)[/tex]:
[tex]\[ 2a - b = 2(4k) - 5k = 8k - 5k = 3k \][/tex]
Now we can write the ratio [tex]\( a + b : 2a - b \)[/tex]:
[tex]\[ a + b : 2a - b = 9k : 3k \][/tex]
Since [tex]\( k \)[/tex] is a common factor, we can simplify the ratio by dividing both terms by [tex]\( k \)[/tex]:
[tex]\[ a + b : 2a - b = 9 : 3 \][/tex]
Thus, the required ratio [tex]\( a + b : 2a - b \)[/tex] is [tex]\( 9 : 3 \)[/tex].
