Understanding Proportion

Proportion means that two ratios are equal. If four numbers $a, b, c, d$ are in proportion, it means that the ratio of the first two numbers is equal to the ratio of the last two numbers. Mathematically, this is expressed as:

$ \frac{a}{b} = \frac{c}{d} $

In this problem, we are given the numbers $94, 24, 100,$ and $26$. We need to find a single number, let's call it '$x$', which, when added to each of these numbers, makes the new set of numbers ($94+x, 24+x, 100+x, 26+x$) proportional.

Setting up the Proportion Problem

Let the number to be added be '$x$'. According to the problem, the new numbers should be in proportion. So, we can write the proportion as:

$ \frac{94+x}{24+x} = \frac{100+x}{26+x} $

Algebraic Calculation

To find the value of '$x$', we need to solve this equation. We can do this by cross-multiplying:

$ (94+x)(26+x) = (24+x)(100+x) $

Now, expand both sides of the equation using the distributive property (or FOIL method):

Left side:

$ (94 \times 26) + (94 \times x) + (x \times 26) + (x \times x) $ $ 2444 + 94x + 26x + x^2 $ $ 2444 + 120x + x^2 $

Right side:

$ (24 \times 100) + (24 \times x) + (x \times 100) + (x \times x) $ $ 2400 + 24x + 100x + x^2 $ $ 2400 + 124x + x^2 $

Now, set the expanded expressions equal to each other:

$ 2444 + 120x + x^2 = 2400 + 124x + x^2 $

We can subtract $x^2$ from both sides, as it appears on both sides:

$ 2444 + 120x = 2400 + 124x $

Now, rearrange the terms to group the constants on one side and the '$x$' terms on the other. Subtract $120x$ from both sides:

$ 2444 = 2400 + 124x - 120x $ $ 2444 = 2400 + 4x $

Subtract $2400$ from both sides:

$ 2444 - 2400 = 4x $ $ 44 = 4x $

Finally, divide by 4 to find the value of '$x$':

$ x = \frac{44}{4} $ $ x = 11 $

Verification

Let's check if adding $11$ to each number results in a proportion:

• $94 + 11 = 105$
• $24 + 11 = 35$
• $100 + 11 = 111$
• $26 + 11 = 37$

Now, let's check if the ratios are equal:

First ratio: $ \frac{105}{35} $

Second ratio: $ \frac{111}{37} $

Calculating the values:

• $ \frac{105}{35} = 3 $
• $ \frac{111}{37} = 3 $

Since both ratios are equal to $3$, the numbers $105, 35, 111,$ and $37$ are in proportion. Therefore, the number that should be added is $11$.