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In this formula, $I$ is the amount of interest, $P$ is the principal, $r$ is the annual interest rate, and $t$ is the time in years. Since Vincenzo's money will stay in this account for $10$ years, substitute the values $P=5000,$ $r=4\%,$ and $t=10$ into the formula to calculate the total amount of interest. The account will earn $\$2000$ in $10$ years. As a final step, add this amount to the principal to calculate the final balance in $10$ years. The balance of the account will be $\$7000$ after $10$ years. Substitute these values into the formula and solve for the annual interest rate $r.$ The bank offers a $0.5\%$ annual interest rate.

It is given that the sales tax rate in Miami is $7\%.$ This means that to purchase an item, an additional $7\%$ of its original price must be paid as sales and use tax. Calculate the tax for a $\$120$ coat in two ways to determine who is correct, Vincenzo or the salesperson.

Vincenzo's Way

Vincenzo prefers to calculate the total cost in two steps, by finding $7\%$ of $\$120$ and then adding this amount to the original price of the coat. Recall that a percent of a number can be calculated as a product of the percent and that number. The percent equation can be helpful to do that. In this equation, $a$ represents the amount of tax, $p$ is the percent, and $w$ is the original price of the item. Substitute the values into the percent equation and calculate $7\%$ of $\$120.$ Remember to write the percent as a decimal number. The amount of the tax for a $\$120$ coat is $\$8.40.$ Now add this amount to the original price of the coat to find the total cost. Vincenzo can buy this coat for $\$128.40.$ Next, calculate the total cost following the salseperson's method.

The salesperson's Way

The salesperson calculates the total cost by multiplying $\$120$ by $1.07.$ Check the result to see if it is the same as Vincenzo's solution. According to the salesperson's calculations, the total cost of the coat is $\$128.40.$

Conclusion

Notice that both methods gave the same result. This is because multiplying $120$ by $7\%,$ or $\frac{7}{100},$ and adding the result to $120$ is the same as multiplying $120$ by $1.07.$ Both of the ways are correct and give the same total cost. This means that both Vincenzo and the salesperson are correct.

According to the receipt, the jeans cost $\$34.00$ and Vincenzo received a discount of $\$10.00.$ Now, recall the formula for the percent equation. In the percent equation, the part of the whole $a$ is the percent $p$ multiplied by the whole $w.$ Here, the discount of $10.00$ dollars is the part $a,$ the percent is $p,$ and the whole $w$ is the original price, $34.00$ dollars. The discount is equal to about $29\%$ of the original price. The in-store price of the sneakers is $120\%$ of the price online. Now, recall the formula for the percent equation once again. According to the receipt, the in-store price of the sneakers is $\$44.00.$ This represents the part $a.$ The percent is $120$ and the whole $w$ is the price online. Now, solve the equation for $w.$ The price of the sneakers is about $\$36.67$ on the online platform.

Start by calculating the total value of the contract. Since Vincenzo's aunt earns $\$1200$ per month, multiply this amount by $24$ to find the total amount of money that she earns in two years. She earns $28800$ dollars in two years. Per her contract, $8\%$ of this amount will be given as a bonus. The percent equation can be used to calculate $8\%$ of $28800.$ Recall that the part $a$ is equal to the product of the percent $p$ and the whole $w.$ Substitute $p\%=8\%$ and $w=28800$ into the percent equation to find the value of $a.$ Vincenzo's aunt will get $\$2304$ as a bonus after working for the company for two years.

Vincenzo's uncle wants to know how much money he will pay in interest. Start by using the percent equation to find the amount of interest he will pay in one year. In this case, the $\text{principal}$ of the loan is $\$2400$ and the $\text{interest}$ $\text{rate}$ is $3.5\%.$ Let the $\text{interest}$ $\text{amount}$ be represented by $s.$ For simplicity, ignore the units for now. Next, solve the equation for $s.$ Remember that percents can be written as decimals by moving the decimal point two places to the left. This means that the simple interest for one year is equal to $\$84.$ Finally, multiply the interest for one year by $4$ to calculate the total amount of interest that Vincenzo's uncle will pay over $4$ years. Vincenzo's uncle will pay $\$336$ in total in interest. It is worth keeping in mind that this amount can be found directly by using the simple interest formula. In this formula, $I$ is the amount of interest, $P$ is the principal, $r$ is the interest rate, and $t$ is the time in years. This means that he will pay $\$2736$ altogether when repaying the loan.