To determine the ratio [tex]\( c:g:t \)[/tex] from the given ratios [tex]\( c:g = 2:3 \)[/tex] and [tex]\( g:t = 5:4 \)[/tex], we need to find a common term for [tex]\( g \)[/tex] in both given ratios. This step-by-step process helps us achieve that:
1. Establish a common term for [tex]\( g \)[/tex]:
- From the ratio [tex]\( c:g = 2:3 \)[/tex], we see that [tex]\( g \)[/tex] is 3 parts.
- From the ratio [tex]\( g:t = 5:4 \)[/tex], we see that [tex]\( g \)[/tex] is 5 parts.
- To make [tex]\( g \)[/tex] the same in both ratios, we need to find a common multiple of 3 and 5.
2. Finding the Least Common Multiple (LCM) of 3 and 5:
- The LCM of 3 and 5 is 15.
3. Adjust the ratios to reflect [tex]\( g \)[/tex] as the common term:
- For the ratio [tex]\( c:g = 2:3 \)[/tex], multiply each part by 5 (since [tex]\( 15 / 3 = 5 \)[/tex]):
- [tex]\( c \)[/tex] becomes [tex]\( 2 \times 5 = 10 \)[/tex]
- [tex]\( g \)[/tex] becomes [tex]\( 3 \times 5 = 15 \)[/tex]
- So, the ratio [tex]\( c:g \)[/tex] becomes [tex]\( 10:15 \)[/tex].
- For the ratio [tex]\( g:t = 5:4 \)[/tex], multiply each part by 3 (since [tex]\( 15 / 5 = 3 \)[/tex]):
- [tex]\( g \)[/tex] becomes [tex]\( 5 \times 3 = 15 \)[/tex]
- [tex]\( t \)[/tex] becomes [tex]\( 4 \times 3 = 12 \)[/tex]
- So, the ratio [tex]\( g:t \)[/tex] becomes [tex]\( 15:12 \)[/tex].
4. Combine the adjusted ratios:
- Now we have [tex]\( c:g = 10:15 \)[/tex] and [tex]\( g:t = 15:12 \)[/tex], with [tex]\( g \)[/tex] being 15 in both.
- This allows us to write the combined ratio [tex]\( c:g:t = 10:15:12 \)[/tex].
Hence, the simplest form of the ratio [tex]\( c:g:t \)[/tex] is [tex]\( 10:15:12 \)[/tex].
