Algebra

An amount of money invested for 1 year in a savings account will earn $400. That same amount of money invested in a mini-mall development will earn $2,400 interest in a year, because the interest paid is 10% more than that paid by the savings account. Find the rate of interest paid by each investment (as percents).

Step-by-step solution with explanation

Final Answer

Savings account rate: ; Mini-mall rate:

Step-by-step solution

1

Set up variables for interest rates

We name the unknown. The savings account rate is what we want to find first, and we can express the mini-mall rate in terms of it.
2

Express the mini-mall rate in terms of r

The problem says the mini-mall pays 10% MORE than the savings account. That means we add 0.10 (which is 10%) to the savings rate r.
3

Write the interest formula for each investment

Simple interest is I = Prt. Since t = 1 year, it simplifies to I = P·r. We write one equation for each investment using the same principal P.
4

Solve the first equation for P

We isolate P by dividing both sides of 400 = P·r by r. This lets us substitute into the second equation.
5

Substitute P into the second equation

We replace P in the mini-mall equation with 400/r. Now we have one equation with one unknown, r.
6

Distribute and simplify

Multiply out: (400/r)·r = 400 and (400/r)·0.10 = 40/r. Then subtract 400 from both sides to get 2000 = 40/r.
7

Solve for r, then find both rates

Divide both sides by 2000 to get r = 0.02, which is 2%. The mini-mall rate is 2% + 10% = 12%.

Understanding this problem

Learning Insight

This problem uses the simple interest formula I = Prt with t = 1. The key idea is that both investments use the SAME principal P — that connection lets us write two equations and link them together to find the unknown rate.

Quick Tip

Whenever two investments share the same principal, divide one interest equation by the other to eliminate P instantly: 2400/400 = (r+0.10)/r gives 6 = (r+0.10)/r, which also solves quickly.

Common Mistake

Students often read '10% more' as multiplying by 1.10, writing the mini-mall rate as 1.10r instead of r + 0.10. '10% more interest RATE' means adding 0.10, not scaling by 1.10.