# Mr. Gonzales invested 10,000 more than her husband did. If they both invested the money at 5 percent per year & their combined income from their investments was 4,000, how much did they invest each?

Sep 9, 2017

The husband: 35,000
The wife: 45,000

#### Explanation:

• Let us start by finding the total value invested which can be calculated from the income of their investments, since we know that the income (4,000) is 5% of the total investment:

("Total value invested")*5%=4,000
$\left(\text{Total value invested}\right) \cdot \frac{5}{100} = 4 , 000$

To isolate the total value invested on the right side, we have to divide 4,000 by 5% (which is multiplying by the inverse of 5% (multiplying by $\frac{100}{5} = 20$)),

we multiply both sides of the equation by 20:
$\left(\text{Total value invested}\right) \cdot \frac{5}{100} \cdot \frac{100}{5} = 4 , 000 \cdot \frac{100}{5} = 4 , 000 \cdot 20$

$\frac{5}{100}$ and $\frac{100}{5}$ will cancel out, and we end with:
$\left(\text{Total value invested}\right) = 4 , 000 \cdot 20 = 80 , 000$

• We continue by assigning the value of what the husband gave to $x$, and the value of what the wife invested to $y$:
• $x = \text{husband's investment}$
• $y = \text{wife's investment}$

Now let us restate find the total value invested in terms of x and y:

$\text{Total value invested} = x + y$

Now since we know that the wife invested 10,000 more than her husband, we can safely say that her investment is 10,000 more than her husband's investments, or in other way: (finding $y$ in terms of $x$)

$y = x + 10 , 000$

Now we can substitute the $y$ by $x + 10 , 000$ in the (total value invested)'s equation:

$\text{Total value invested} = x + y = x + \left(x + 10000\right) = 2 x + 10 , 000$

Now we put the value of total value invested in the equation and solve for $x$
$\text{Total value invested} = 2 x + 10 , 000$
$80 , 000 = 2 x + 10 , 000$

Subtracting both sides of the equation by 10,000 to isolate the unknown's (x's) term:

$\rightarrow 80 , 000 - 10 , 000 = 2 x + 10 , 000 - 10 , 000$
$= 70 , 000 = 2 x$

Dividing both sides of the equation by 2 to isolate x:

$\rightarrow 70 , \frac{000}{2} = 2 \frac{x}{2}$
$35 , 000 = x = \text{husband's investment}$

Wife's investment is the husband's investment $+ 10 , 000$ which is equal to:
$35 , 000 + 10 , 000 = 45 , 000$