Understanding and Solving Proportions

The question asks us to find the value of N in the given proportion: N : 38 ∷ 3 : 57. This notation means that the ratio N to 38 is equal to the ratio 3 to 57.

What is a Proportion?

A proportion is a statement that two ratios are equal. In general, a proportion can be written as a : b ∷ c : d, which is the same as saying $\frac{a}{b} = \frac{c}{d}$. Here, 'a' and 'd' are called the 'extremes', and 'b' and 'c' are called the 'means'. A key property of proportions is that the product of the means equals the product of the extremes (b * c = a * d).

Setting up the Proportion Equation

Based on the definition of proportion, we can write the given relationship as an equation:

$\frac{N}{38} = \frac{3}{57}$

Solving for N using the Equation

To find the value of N, we need to isolate N in the equation. We can do this by multiplying both sides of the equation by 38:

$N = \frac{3}{57} \times 38$

Now, let's simplify the fraction $\frac{3}{57}$. We can see that both 3 and 57 are divisible by 3:

$3 \div 3 = 1$

$57 \div 3 = 19$

So, the fraction simplifies to $\frac{1}{19}$.

Substitute this simplified fraction back into the equation for N:

$N = \frac{1}{19} \times 38$

Now, multiply $\frac{1}{19}$ by 38:

$N = \frac{38}{19}$

Performing the division:

$38 \div 19 = 2$

So, $N = 2$.

Alternative Method: Using the Property of Proportion

We could also use the property that the product of the means equals the product of the extremes. In the proportion N : 38 ∷ 3 : 57:

• Extremes are N and 57.
• Means are 38 and 3.

So, Product of Extremes = Product of Means

$N \times 57 = 38 \times 3$

$57N = 114$

To find N, divide both sides by 57:

$N = \frac{114}{57}$

Performing the division:

$114 \div 57 = 2$

So, $N = 2$. Both methods give the same result.

Verification with Options

We found that N = 2. Let's look at the provided options:

1. 2/3
2. 1/3
3. 3
4. 2

Our calculated value N = 2 matches option 4.

Summary of Solution Steps

1. Interpret the proportion notation as an equality of ratios: $\frac{N}{38} = \frac{3}{57}$.
2. Solve the equation for N, either by multiplying both sides by 38 or by using the product of means and extremes property.
3. Simplify calculations to find the value of N.
4. Match the calculated value of N with the given options.

Revision Table: Key Proportion Concepts

Additional Information: Properties of Proportions

Proportions have several useful properties that can help in solving problems:

• Cross-Multiplication Property: If $\frac{a}{b} = \frac{c}{d}$, then $ad = bc$. This is the basis for the product of means and extremes.
• Invertendo: If $\frac{a}{b} = \frac{c}{d}$, then $\frac{b}{a} = \frac{d}{c}$ (provided a and c are not zero).
• Alternendo: If $\frac{a}{b} = \frac{c}{d}$, then $\frac{a}{c} = \frac{b}{d}$ (provided c and b are not zero).
• Componendo: If $\frac{a}{b} = \frac{c}{d}$, then $\frac{a+b}{b} = \frac{c+d}{d}$.
• Dividendo: If $\frac{a}{b} = \frac{c}{d}$, then $\frac{a-b}{b} = \frac{c-d}{d}$.
• Componendo and Dividendo: If $\frac{a}{b} = \frac{c}{d}$, then $\frac{a+b}{a-b} = \frac{c+d}{c-d}$ (provided $a \neq b$ and $c \neq d$).

These properties can be useful for manipulating and solving more complex proportion problems.