If (a + b + c): d = 5: 1, (b + c + d): a = 7: 1 and, (c + d + a): b = 3: 1 then find a: b: c: d?
3 : 6 : 11 : 4
Understanding the Ratio and Proportion Problem
This problem asks us to find the ratio of four variables, \(a:b:c:d\), given three proportional relationships involving sums of these variables. We are given:
- \((a + b + c): d = 5: 1\)
- \((b + c + d): a = 7: 1\)
- \((c + d + a): b = 3: 1\)
To find the ratio \(a:b:c:d\), we need to express each variable in terms of a common value or find their relative values.
Step-by-Step Solution to Find the Ratio
Let's convert the given ratios into equations:
- From \((a + b + c): d = 5: 1\), we have \(\frac{a + b + c}{d} = \frac{5}{1}\). This gives us the equation: \(a + b + c = 5d\)
- From \((b + c + d): a = 7: 1\), we have \(\frac{b + c + d}{a} = \frac{7}{1}\). This gives us the equation: \(b + c + d = 7a\)
- From \((c + d + a): b = 3: 1\), we have \(\frac{c + d + a}{b} = \frac{3}{1}\). This gives us the equation: \(c + d + a = 3b\)
Notice that the sum of all four variables \(a + b + c + d\) is present in different combinations in these equations. Let's denote the sum \(a + b + c + d\) as \(S\).
We can rewrite the equations by adding the missing variable to both sides to get the total sum \(S\):
- For the first equation, \(a + b + c = 5d\), add \(d\) to both sides: \((a + b + c) + d = 5d + d \implies S = 6d\)
- For the second equation, \(b + c + d = 7a\), add \(a\) to both sides: \((b + c + d) + a = 7a + a \implies S = 8a\)
- For the third equation, \(c + d + a = 3b\), add \(b\) to both sides: \((c + d + a) + b = 3b + b \implies S = 4b\)
Now we have expressed \(a\), \(b\), and \(d\) in terms of \(S\):
- \(S = 6d \implies d = \frac{S}{6}\)
- \(S = 8a \implies a = \frac{S}{8}\)
- \(S = 4b \implies b = \frac{S}{4}\)
We know that \(S = a + b + c + d\). We can substitute the expressions for \(a\), \(b\), and \(d\) into this equation to find \(c\) in terms of \(S\):
\(S = \frac{S}{8} + \frac{S}{4} + c + \frac{S}{6}\)
To solve for \(c\), isolate \(c\) on one side:
\(c = S - \frac{S}{8} - \frac{S}{4} - \frac{S}{6}\)
Find a common denominator for 8, 4, and 6. The least common multiple (LCM) of 8, 4, and 6 is 24.
\(c = \frac{24S}{24} - \frac{3S}{24} - \frac{6S}{24} - \frac{4S}{24}\)
Combine the terms:
\(c = \frac{24S - 3S - 6S - 4S}{24}\)
\(c = \frac{(24 - 3 - 6 - 4)S}{24}\)
\(c = \frac{11S}{24}\)
Now we have all four variables expressed in terms of \(S\):
- \(a = \frac{S}{8}\)
- \(b = \frac{S}{4}\)
- \(c = \frac{11S}{24}\)
- \(d = \frac{S}{6}\)
To find the ratio \(a : b : c : d\), we write:
\(a : b : c : d = \frac{S}{8} : \frac{S}{4} : \frac{11S}{24} : \frac{S}{6}\)
To simplify this ratio and express it using integers, we multiply each term by the LCM of the denominators (8, 4, 24, 6), which is 24:
\(a : b : c : d = \left(\frac{S}{8} \times 24\right) : \left(\frac{S}{4} \times 24\right) : \left(\frac{11S}{24} \times 24\right) : \left(\frac{S}{6} \times 24\right)\)
\(a : b : c : d = 3S : 6S : 11S : 4S\)
Assuming \(S \ne 0\), we can divide all terms by \(S\):
\(a : b : c : d = 3 : 6 : 11 : 4\)
This gives us the final ratio for \(a:b:c:d\).
| Given Ratio | Equation | Sum \(S = a+b+c+d\) | Variable in terms of \(S\) |
|---|---|---|---|
| (a+b+c) : d = 5 : 1 | a+b+c = 5d | S = 6d | \(d = S/6\) |
| (b+c+d) : a = 7 : 1 | b+c+d = 7a | S = 8a | \(a = S/8\) |
| (c+d+a) : b = 3 : 1 | c+d+a = 3b | S = 4b | \(b = S/4\) |
| Derived value for c | c = S - a - b - d | \(S = a+b+c+d\) | \(c = 11S/24\) |
Ratio a:b:c:d Final Result
Based on our calculations, the ratio \(a:b:c:d\) is \(3:6:11:4\).
Revision Table for Ratio Problem
| Initial Information | Derived Information | Final Ratio |
|---|---|---|
| (a+b+c):d = 5:1 | S = 6d | a : b : c : d = 3 : 6 : 11 : 4 |
| (b+c+d):a = 7:1 | S = 8a | |
| (c+d+a):b = 3:1 | S = 4b | |
| S = a+b+c+d | \(a=S/8, b=S/4, c=11S/24, d=S/6\) |
Additional Information on Ratio and Proportion
A ratio is a comparison of two quantities by division. For example, the ratio \(a:b\) means \(\frac{a}{b}\). A proportion is an equation stating that two ratios are equal.
In this problem, we used the property that if a set of quantities (\(a, b, c, d\)) have a constant sum \(S\), and parts of the sum are in proportion to one of the quantities, we can express each quantity relative to the total sum \(S\).
If \((a+b+c)/d = k\), this means \(a+b+c = kd\). Adding \(d\) to both sides gives \(a+b+c+d = kd+d = (k+1)d\). If we let \(S = a+b+c+d\), then \(S = (k+1)d\), so \(d = S/(k+1)\). We applied this concept for each given ratio to express \(a, b, c, d\) in terms of \(S\).
Once each variable is expressed as a fraction of the common sum \(S\), finding the ratio involves comparing these fractions and scaling them up to the smallest possible integers by multiplying by the LCM of their denominators.
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Important Questions from Simple Ratios
If x : y = 5 : 2, then the value of (8x + 9y) : (8x + 2y) is:
If the value of \(\frac{{p - q}}{q} = \frac{4}{3}\) .find p : q
Two numbers are in the ratio of 9 : 7. If the larger number is 56 more than one-seventh of the smaller, then what is the sum of the two numbers?
The ratio of two numbers A and B is 5: 8. If 5 is added to each of A and B, then the ratio becomes 2 : 3. The difference between A and B is:
In population of a city the ratio of men and women is 12 : 11. If total population of that city is 4,60,000, then the population of women is:
