Of course! Let's break down each problem step-by-step to find the required numbers.
### Part (a)
We are given:
- The ratio of two numbers is [tex]\( 4 : 5 \)[/tex].
- The first number is 20.
Let's call the first number [tex]\( x \)[/tex] and the second number [tex]\( y \)[/tex]. So, we have:
[tex]\[ \frac{x}{y} = \frac{4}{5} \][/tex]
We know:
[tex]\[ x = 20 \][/tex]
We need to find [tex]\( y \)[/tex].
Using the ratio we can write,
[tex]\[ \frac{20}{y} = \frac{4}{5} \][/tex]
Cross-multiplying to solve for [tex]\( y \)[/tex]:
[tex]\[ 20 \cdot 5 = 4 \cdot y \][/tex]
[tex]\[ 100 = 4y \][/tex]
Divide both sides by 4:
[tex]\[ y = \frac{100}{4} \][/tex]
[tex]\[ y = 25 \][/tex]
Thus, the second number is:
[tex]\[ y = 25 \][/tex]
### Part (b)
We are given:
- The ratio of two numbers is [tex]\( 9 : 5 \)[/tex].
- The second number is 45.
Let's call the first number [tex]\( a \)[/tex] and the second number [tex]\( b \)[/tex]. So, we have:
[tex]\[ \frac{a}{b} = \frac{9}{5} \][/tex]
We know:
[tex]\[ b = 45 \][/tex]
We need to find [tex]\( a \)[/tex].
Using the ratio we can write,
[tex]\[ \frac{a}{45} = \frac{9}{5} \][/tex]
Cross-multiplying to solve for [tex]\( a \)[/tex]:
[tex]\[ a \cdot 5 = 9 \cdot 45 \][/tex]
[tex]\[ 5a = 405 \][/tex]
Divide both sides by 5:
[tex]\[ a = \frac{405}{5} \][/tex]
[tex]\[ a = 81 \][/tex]
Thus, the first number is:
[tex]\[ a = 81 \][/tex]
Final answers:
- For part (a), the second number is 25.
- For part (b), the first number is 81.
