To solve the problem where [tex]\(a : b = 8 : 5\)[/tex] and we need to find [tex]\((7a + 5b) : (8a - 9b)\)[/tex], follow these steps:
1. Given [tex]\(a : b = 8 : 5\)[/tex], it implies that [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are in the ratio 8 to 5. We can represent this by [tex]\(a = 8k\)[/tex] and [tex]\(b = 5k\)[/tex] for some constant [tex]\(k\)[/tex].
2. Express [tex]\(7a + 5b\)[/tex]:
[tex]\[
7a + 5b = 7(8k) + 5(5k)
\][/tex]
Simplify the expression:
[tex]\[
7a + 5b = 56k + 25k = 81k
\][/tex]
3. Express [tex]\(8a - 9b\)[/tex]:
[tex]\[
8a - 9b = 8(8k) - 9(5k)
\][/tex]
Simplify the expression:
[tex]\[
8a - 9b = 64k - 45k = 19k
\][/tex]
4. Form the ratio [tex]\((7a + 5b) : (8a - 9b)\)[/tex]:
Using the simplified forms:
[tex]\[
(7a + 5b) : (8a - 9b) = 81k : 19k
\][/tex]
Since [tex]\(k\)[/tex] is a common factor, it cancels out:
[tex]\[
= 81 : 19
\][/tex]
5. Calculate Numerical Values:
To represent this ratio as a division and get a decimal form:
[tex]\[
\frac{(7a + 5b)}{(8a - 9b)} = \frac{81k}{19k} = \frac{81}{19} \approx 4.2631578947368425
\][/tex]
6. Conclusion:
The ratio [tex]\((7a + 5b) : (8a - 9b)\)[/tex] is [tex]\(81 : 19\)[/tex]. When expressed as a decimal, this ratio is approximately [tex]\(4.2631578947368425\)[/tex].
Thus, the detailed solution finds that the simplified ratio is [tex]\((81, 19)\)[/tex] and the precise decimal value of the ratio is approximately [tex]\(4.2631578947368425\)[/tex].
