If (5a – 3b) : (4a – 2b) = 2 : 3, then a : b is equal to:
5 : 7
The problem asks us to find the ratio a : b given the ratio equation (5a – 3b) : (4a – 2b) = 2 : 3. This involves solving an algebraic equation derived from the given ratio.
We can write the given ratio as a fraction:
$$\frac{5a - 3b}{4a - 2b} = \frac{2}{3}$$
To solve for the relationship between a and b, we can cross-multiply. This means multiplying the numerator of the left side by the denominator of the right side and setting it equal to the product of the denominator of the left side and the numerator of the right side.
Cross-multiplying gives us:
$$3 \times (5a - 3b) = 2 \times (4a - 2b)$$
Now, we distribute the numbers on both sides of the equation:
$$3 \times 5a - 3 \times 3b = 2 \times 4a - 2 \times 2b$$
$$15a - 9b = 8a - 4b$$
Next, we need to group the terms involving a on one side of the equation and the terms involving b on the other side. Let's move the 8a term to the left side by subtracting 8a from both sides, and move the -9b term to the right side by adding 9b to both sides:
$$15a - 8a = -4b + 9b$$
Now, combine the like terms:
$$(15 - 8)a = (-4 + 9)b$$
$$7a = 5b$$
We are asked to find the ratio a : b. This means we want to express the equation 7a = 5b in the form $\frac{a}{b}$. To do this, we can divide both sides of the equation by b (assuming $b \neq 0$) and then divide both sides by 7:
$$\frac{7a}{b} = \frac{5b}{b}$$
$$\frac{7a}{b} = 5$$
Now, divide both sides by 7:
$$\frac{7a}{7b} = \frac{5}{7}$$
$$\frac{a}{b} = \frac{5}{7}$$
The ratio a : b is therefore 5 : 7.
Let's verify with an example. If $a=5k$ and $b=7k$ for some non-zero $k$.
Substitute these into the original ratio:
$$(5a - 3b) : (4a - 2b) = (5(5k) - 3(7k)) : (4(5k) - 2(7k))$$
$$= (25k - 21k) : (20k - 14k)$$
$$= 4k : 6k$$
$$= 4 : 6$$
Dividing both parts of the ratio by 2:
$$= 2 : 3$$
This matches the given ratio, confirming that a : b = 5 : 7 is correct.
| Step | Description | Equation |
|---|---|---|
| 1 | Write ratio as fraction | $$\frac{5a - 3b}{4a - 2b} = \frac{2}{3}$$ |
| 2 | Cross-multiply | $$3(5a - 3b) = 2(4a - 2b)$$ |
| 3 | Distribute | $$15a - 9b = 8a - 4b$$ |
| 4 | Group 'a' and 'b' terms | $$15a - 8a = 9b - 4b$$ |
| 5 | Combine like terms | $$7a = 5b$$ |
| 6 | Express as ratio a:b | $$\frac{a}{b} = \frac{5}{7} \Rightarrow a : b = 5 : 7$$ |
Revision Table: Ratios and Algebra
| Concept | Description | Example |
|---|---|---|
| Ratio | Comparison of two quantities by division. Written as a:b or a/b. | If a and b are in ratio 2:1, then a/b = 2/1. |
| Proportion | An equation stating that two ratios are equal. | a:b = c:d or a/b = c/d. |
| Cross-Multiplication | Method to solve equations involving proportions. For a/b = c/d, ad = bc. | Given x/4 = 3/12, 12x = 4*3 = 12. So x=1. |
| Solving Linear Equations | Rearranging terms to isolate the variable. | Given 2x + 3 = 7, 2x = 7-3 = 4, x = 4/2 = 2. |
Additional Information: Algebraic Ratios
When dealing with ratios involving algebraic expressions like (5a - 3b) : (4a - 2b), treating the ratio as a fraction is a standard method for solving. The principle is that if the ratio of two quantities is equal to the ratio of two other quantities, they form a proportion.
A ratio x : y can always be written as a fraction $\frac{x}{y}$. So, (5a - 3b) : (4a - 2b) = 2 : 3 directly translates to the equation $\frac{5a - 3b}{4a - 2b} = \frac{2}{3}$.
Cross-multiplication is a powerful technique for solving such fractional equations. It converts the equation from a fractional form into a linear equation, which is generally easier to solve. Always remember to distribute correctly when you multiply a number or variable into a parenthesis.
Finally, expressing the relationship 7a = 5b as a ratio a : b requires isolating the $\frac{a}{b}$ term. This is done by dividing both sides of the equation appropriately. The result $\frac{a}{b} = \frac{5}{7}$ directly gives the ratio a : b as 5 : 7.
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