To solve the problem of finding the ratio \( a:b:c:d \) given the ratios \( a:b = \frac{2}{3}:\frac{1}{3} \), \( b:c = \frac{2}{7}:\frac{5}{14} \), and \( d:c = \frac{7}{10}:\frac{3}{5} \), we can follow these steps:
### Step 1: Simplify \( a:b \)
Given \( a:b = \frac{2}{3}:\frac{1}{3} \), we can express this as:
\[
a:b = 2:1
\]
This means for every 2 parts of \( a \), there is 1 part of \( b \).
**Hint:** To simplify a ratio, divide both parts by the same number or multiply to eliminate fractions.
### Step 2: Simplify \( b:c \)
Given \( b:c = \frac{2}{7}:\frac{5}{14} \), we can eliminate the fractions by multiplying both sides by 14 (the least common multiple of the denominators):
\[
b:c = 4:5
\]
This means for every 4 parts of \( b \), there are 5 parts of \( c \).
**Hint:** To eliminate fractions in a ratio, multiply by the least common multiple of the denominators.
### Step 3: Simplify \( d:c \)
Given \( d:c = \frac{7}{10}:\frac{3}{5} \), we can eliminate the fractions by multiplying both sides by 10:
\[
d:c = 7:6
\]
This means for every 7 parts of \( d \), there are 6 parts of \( c \).
**Hint:** Again, multiply by the least common multiple to simplify ratios with fractions.
### Step 4: Express all ratios in terms of a common variable
From \( a:b = 2:1 \), we can express \( a \) and \( b \) as:
\[
a = 2x, \quad b = x
\]
From \( b:c = 4:5 \), we can express \( b \) and \( c \) as:
\[
b = 4y, \quad c = 5y
\]
From \( d:c = 7:6 \), we can express \( d \) and \( c \) as:
\[
d = 7z, \quad c = 6z
\]
### Step 5: Equate \( b \) and \( c \)
Since \( b \) appears in both \( a:b \) and \( b:c \), we can set \( x = 4y \):
\[
x = 4y \implies b = 4y
\]
Substituting \( b \) into \( a \):
\[
a = 2(4y) = 8y
\]
Now, equate \( c \) from \( b:c \) and \( d:c \):
From \( b:c \):
\[
c = 5y
\]
From \( d:c \):
\[
c = 6z
\]
Setting \( 5y = 6z \) gives:
\[
y = \frac{6}{5}z
\]
### Step 6: Substitute back to find \( a, b, c, d \)
Substituting \( y \) back into \( a, b, c, d \):
- \( a = 8y = 8 \times \frac{6}{5}z = \frac{48}{5}z \)
- \( b = 4y = 4 \times \frac{6}{5}z = \frac{24}{5}z \)
- \( c = 5y = 5 \times \frac{6}{5}z = 6z \)
- \( d = 7z \)
### Step 7: Find a common multiple
To express \( a:b:c:d \) in whole numbers, we can multiply by 5:
\[
a = 48, \quad b = 24, \quad c = 30, \quad d = 35
\]
### Final Ratio
Thus, the final ratio \( a:b:c:d \) is:
\[
\boxed{48:24:30:35}
\]
