Samson opened a bank account with $$ 1.25 \% $$ simple interest. The total amount of interest Samson will earn after 20 years is $$ \$ 812.50 $$. How much did Samson deposit when he opened the account? Which equation represents the total interest, $$ T $$, earned when the principal amount is $$ \$ 100 $$, the annual simple interest rate is $$ 1 \% $$, and the number of years is 10 ? (A) $$ T=100 \cdot(10+0.1) $$ (C) $$ T=100 \cdot 0.1 \cdot 10 $$ (B) $$ T=100 \cdot(10+0.01) $$ (D) $$ T=100 \cdot 0.01 \cdot 10 $$ 3. Mika opened a bank account that earns simple interest with an initial deposit of $$ \$ 2,200 $$. She made no other transactions throughout the year. At the end of the year, Mika had $$ \$ 2,260.50 $$ in her account. What was the simple interest rate? 4. Vince borrows $$ \$ 900 $$ to buy a couch. He will pay off the loan by paying $$ 1.5 \% $$ simple interest for 2 years. Vince incorrectly calculates the amount he will pay back using the expression below. $$ 900+900(1.015 \cdot 2) $$ What is the correct amount Vince will pay back altogether? Explain the error in Vince's expression. 5. Darryl deposits $$ \$ 1,500 $$ into a savings account that has a simple interest rate of $$ 2.7 \% $$. Lori deposits $$ \$ 1,400 $$ into a savings account that has a simple interest rate of $$ 3.8 \% $$. If no other transactions are made, who will have more money in their account after 10 years? How much more? (A) After 10 years, Lori will have $$ \$ 27 $$ less than Darryl. (B) After 10 years, Lori will have $$ 1.1 \% $$ more than Darryl. (C) After 10 years, Darryl will have $$ \$ 16.50 $$ more than Lori. (D) After 10 years, Lori will have $$ \$ 27 $$ more than Darryl.

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Let's tackle each of your questions one by one:

---

### **1. Samson's Initial Deposit**

**Problem Statement:**
Samson opened a bank account with a **1.25%** simple interest rate. The total interest earned after **20 years** is **\$812.50**. Determine the initial deposit.

**Solution:**
The formula for simple interest is:
$$
\text{Interest} (I) = \text{Principal} (P) \times \text{Rate} (r) \times \text{Time} (t)
$$

Given:
- $$ I = \$812.50 $$
- $$ r = 1.25\% = 0.0125 $$
- $$ t = 20 $$ years

Plugging the values into the formula:
$$
812.50 = P \times 0.0125 \times 20
$$
$$
812.50 = P \times 0.25
$$
$$
P = \frac{812.50}{0.25} = \$3,250
$$

**Answer:** Samson initially deposited **\$3,250**.

---

### **2. Selecting the Correct Interest Formula**

**Problem Statement:**
Which equation represents the total interest, $$ T $$, earned when the principal amount is **\$100**, the annual simple interest rate is **1%**, and the number of years is **10**?

**Options:**
- (A) $$ T = 100 \cdot (10 + 0.1) $$
- (B) $$ T = 100 \cdot (10 + 0.01) $$
- (C) $$ T = 100 \cdot 0.1 \cdot 10 $$
- (D) $$ T = 100 \cdot 0.01 \cdot 10 $$

**Solution:**
Using the simple interest formula:
$$
T = P \times r \times t
$$
Where:
- $$ P = \$100 $$
- $$ r = 1\% = 0.01 $$
- $$ t = 10 $$ years

Plugging in the values:
$$
T = 100 \times 0.01 \times 10 = 100 \times 0.1 = \$10
$$

**Option Analysis:**
- **Option (D)** correctly represents the formula:
  $$
  T = 100 \times 0.01 \times 10
  $$

**Answer:** **(D)** $$ T = 100 \cdot 0.01 \cdot 10 $$

---

### **3. Mika's Simple Interest Rate**

**Problem Statement:**
Mika deposited **\$2,200** into an account earning simple interest. After one year, her account balance is **\$2,260.50**. Determine the simple interest rate.

**Solution:**
First, find the interest earned:
$$
I = \$2,260.50 - \$2,200 = \$60.50
$$

Using the simple interest formula:
$$
I = P \times r \times t
$$
Where:
- $$ I = \$60.50 $$
- $$ P = \$2,200 $$
- $$ t = 1 $$ year

Solving for $$ r $$:
$$
60.50 = 2200 \times r \times 1
$$
$$
r = \frac{60.50}{2200} = 0.0275 = 2.75\%
$$

**Answer:** The simple interest rate is **2.75%**.

---

### **4. Correcting Vince's Loan Repayment Calculation**

**Problem Statement:**
Vince borrows **\$900** at a **1.5%** simple interest rate for **2 years**. He incorrectly calculates the total repayment using:
$$
900 + 900(1.015 \times 2)
$$
Determine the correct total amount Vince should pay back and explain the error.

**Solution:**
**Correct Calculation:**
Simple interest formula:
$$
I = P \times r \times t
$$
Where:
- $$ P = \$900 $$
- $$ r = 1.5\% = 0.015 $$
- $$ t = 2 $$ years

Calculating interest:
$$
I = 900 \times 0.015 \times 2 = 900 \times 0.03 = \$27
$$

Total repayment:
$$
900 + 27 = \$927
$$

**Error in Vince's Calculation:**
Vince used the expression $$ 1.015 \times 2 = 2.03 $$, which incorrectly applies compound-like reasoning to a simple interest scenario. The correct approach for simple interest is to multiply only the rate and time, not to compound the rate.

**Answer:** Vince should pay back **\$927** in total. His error was multiplying the principal by $$ (1.015 \times 2) $$ instead of correctly applying the simple interest formula $$ P \times r \times t $$.

---

### **5. Comparing Darryl's and Lori's Savings After 10 Years**

**Problem Statement:**
- **Darryl** deposits **\$1,500** at a **2.7%** simple interest rate.
- **Lori** deposits **\$1,400** at a **3.8%** simple interest rate.
  
After **10 years**, who has more, and by how much?

**Options:**
- (A) After 10 years, Lori will have **\$27** less than Darryl.
- (B) After 10 years, Lori will have **1.1%** more than Darryl.
- (C) After 10 years, Darryl will have **\$16.50** more than Lori.
- (D) After 10 years, Lori will have **\$27** more than Darryl.

**Solution:**
**Darryl's Account:**
$$
I_1 = 1500 \times 0.027 \times 10 = 1500 \times 0.27 = \$405
$$
$$
\text{Total} = 1500 + 405 = \$1,905
$$

**Lori's Account:**
$$
I_2 = 1400 \times 0.038 \times 10 = 1400 \times 0.38 = \$532
$$
$$
\text{Total} = 1400 + 532 = \$1,932
$$

**Difference:**
$$
1932 - 1905 = \$27
$$

**Answer:** After 10 years, **Lori will have \$27 more than Darryl**.

**Correct Option:** **(D)** After 10 years, Lori will have **\$27** more than Darryl.

---
**1. Samson's Initial Deposit:** \$3,250 **2. Correct Interest Formula:** $$ T = 100 \cdot 0.01 \cdot 10 $$ (Option D) **3. Mika's Interest Rate:** 2.75% **4. Vince's Repayment:** \$927 (He should have used $$ P \times r \times t $$ instead of $$ P \times (1 + r \times t) $$) **5. After 10 Years:** Lori has \$27 more than Darryl (Option D)

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