We appreciate your visit to In a flowerbed there are only geraniums and roses The ratio of ladybirds on the geraniums to ladybirds on the roses is initially 2 3. This page offers clear insights and highlights the essential aspects of the topic. Our goal is to provide a helpful and engaging learning experience. Explore the content and find the answers you need!
Answer :
To solve this problem, we need to determine the number of ladybirds initially on the geraniums, given the conditions about their movement and ratios. Let's break it down step-by-step:
1. Define Variables:
- Let [tex]\( g \)[/tex] be the initial number of ladybirds on the geraniums.
- Let [tex]\( r \)[/tex] be the initial number of ladybirds on the roses.
2. Set Up the Initial Ratio:
- We know the ratio of ladybirds on the geraniums to ladybirds on the roses is 2:3. This gives us the equation:
[tex]\[
\frac{g}{r} = \frac{2}{3}
\][/tex]
3. Account for Movement:
- 4 ladybirds move from the roses to the geraniums. After this movement:
- The number of ladybirds on the geraniums becomes [tex]\( g + 4 \)[/tex].
- The number of ladybirds on the roses becomes [tex]\( r - 4 \)[/tex].
4. Set Up the New Ratio:
- After the movement, the new ratio of ladybirds on geraniums to roses is 7:8. This gives us another equation:
[tex]\[
\frac{g + 4}{r - 4} = \frac{7}{8}
\][/tex]
5. Solve the System of Equations:
- From the first equation:
[tex]\[
g = \frac{2}{3}r
\][/tex]
- From the second equation:
[tex]\[
8(g + 4) = 7(r - 4)
\][/tex]
- Substituting [tex]\( g = \frac{2}{3}r \)[/tex] into the second equation helps in solving these equations simultaneously.
Once you solve these equations, you find that there were initially 24 ladybirds on the geraniums.
This detailed process allows us to work through the problem step-by-step based on given ratios and movements, leading us to find that the initial number of ladybirds on the geraniums was 24.
1. Define Variables:
- Let [tex]\( g \)[/tex] be the initial number of ladybirds on the geraniums.
- Let [tex]\( r \)[/tex] be the initial number of ladybirds on the roses.
2. Set Up the Initial Ratio:
- We know the ratio of ladybirds on the geraniums to ladybirds on the roses is 2:3. This gives us the equation:
[tex]\[
\frac{g}{r} = \frac{2}{3}
\][/tex]
3. Account for Movement:
- 4 ladybirds move from the roses to the geraniums. After this movement:
- The number of ladybirds on the geraniums becomes [tex]\( g + 4 \)[/tex].
- The number of ladybirds on the roses becomes [tex]\( r - 4 \)[/tex].
4. Set Up the New Ratio:
- After the movement, the new ratio of ladybirds on geraniums to roses is 7:8. This gives us another equation:
[tex]\[
\frac{g + 4}{r - 4} = \frac{7}{8}
\][/tex]
5. Solve the System of Equations:
- From the first equation:
[tex]\[
g = \frac{2}{3}r
\][/tex]
- From the second equation:
[tex]\[
8(g + 4) = 7(r - 4)
\][/tex]
- Substituting [tex]\( g = \frac{2}{3}r \)[/tex] into the second equation helps in solving these equations simultaneously.
Once you solve these equations, you find that there were initially 24 ladybirds on the geraniums.
This detailed process allows us to work through the problem step-by-step based on given ratios and movements, leading us to find that the initial number of ladybirds on the geraniums was 24.
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