Solve each problem. Simple InterestSimple interest varies jointly as principal and time. If \$1000 invested for 2 yr earned \$70, find the amount of interest earned by \$5000 invested for 5 yr.

Verified step by step guidance

1. Identify the formula for simple interest, which varies jointly as principal (P) and time (T). This can be written as \(I = k \times P \times T\), where \(I\) is the interest and \(k\) is the constant of proportionality.
2. Use the given information to find the constant \(k\). Substitute \(I = 70\), \(P = 1000\), and \(T = 2\) into the formula: \$70 = k \times 1000 \times 2$.
3. Solve for \(k\) by isolating it on one side: \(k = \frac{70}{1000 \times 2}\).
4. Now, use the value of \(k\) to find the interest earned when \(P = 5000\) and \(T = 5\). Substitute these values into the formula: \(I = k \times 5000 \times 5\).
5. Calculate the expression for \(I\) to find the interest earned on the \$5000\( investment over \)5$ years.

Key Concepts

Simple Interest Formula

Simple interest is calculated using the formula I = PRT, where I is the interest earned, P is the principal amount, R is the rate of interest per year, and T is the time in years. This formula helps determine the interest earned over a period without compounding.

Joint Variation

Joint variation means a quantity varies directly as the product of two or more other quantities. In this problem, simple interest varies jointly as principal and time, implying I = k × P × T, where k is a constant of proportionality.

Finding the Constant of Proportionality

To solve joint variation problems, first find the constant k by substituting known values into the equation. Once k is found, use it to calculate the unknown interest for different principal and time values.