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LaShay is a brilliant pianist. One day, she is asked to play at her school concert. She is excited to showcase her talent in front of her peers. She now wants to calculate how long it would take her to play her piece.

Solution

Multiplying the number of songs and the duration of each song gives the total time it will take LaShay to play all songs.

$$\begin{array}{c}\text{Total Time}=\text{Number of Songs}\times \text{Length of Song}\end{array}$$

In this case, LaShay has a total of $8$ songs, each taking an average of $3$ minutes to play.

$$\begin{array}{c}\text{Total Time}=8\times 3\end{array}$$

Note that this is a multiplication of two positive integers. The product of this multiplication can be found by using a number line. Keep in mind that the first factor indicates that $8$ equal intervals of $3$ are needed.

The endpoint is $24.$ This is the product of $8\times 3.$ This means that it will take LaShay $24$ minutes to play her $8$ pieces.

$$\begin{array}{c}\text{Total Time}=8\times 3\\ \Downarrow \\ \text{Total Time}=24\end{array}$$

In the afternoons, LaShay takes a break from her studies and piano lessons to help her aunt with her pizza shop.

LaShay's aunt pays her $\$10$ for each day she helps her. However, every time LaShay is late, she loses $\$3.$

Solution

a A loss means something negative. This means that the opposite of $3$ represents the amount of money LaShay loses each day for being late.

$$\begin{array}{c}\text{Amount Lost for Being Late Each Day}\\ \text{-}\$3\end{array}$$

Next, remember that she was late $6$ out of the $15$ days she helped her aunt last month. The amount she lost last month is given by multiplying the number of days she was late by the amount she lost each day for being late.

$$\begin{array}{c}\text{Total Amount Lost Last Month}\\ 6\times (\text{-}3)\end{array}$$

This situation is a multiplication of a positive integer by a negative integer. Recall that the result is negative when multiplying two integers with different signs. With this in mind, perform the multiplication as if both were whole numbers by ignoring their signs.

$$\begin{array}{c}6\times (\text{-}3)\Rightarrow 6\times 3\end{array}$$

The resulting multiplication indicates $6$ equal intervals of $3.$ A number line can be used to help find this product.

The product of the resulting multiplication is $18.$ Finally, change $18$ to its opposite $\text{-}18$ to get the final product of the initial multiplication.

$$\begin{array}{c}6\times 3=18\Rightarrow 6\times (\text{-}3)=\text{-}18\end{array}$$

The amount LaShay lost last month is then $\text{-}\$18.$

b LaShay's total possible earnings are the product of the number of days worked by the amount paid per day gives her total earnings. However, since she was late six days, the amount she lost for being late must be added to the total amount of money she could earn. Start by finding the total possible earnings.

$$\begin{array}{c}\text{Total Possible Earnings}\\ \$10\cdot 15\end{array}$$

This is a multiplication of two positive integers, so the result is also positive. Because the integers are two-digit numbers, use the multiplication digit by digit to find their product.

LaShay could have earned a total of $\$150.$ Add to this amount the amount she lost — $\text{-}\$18$ — to find her total earnings.

$$\begin{array}{c}150+(\text{-}18)\end{array}$$

The result is the sum of a positive and a negative integer. Change the addition sign to a subtraction sign and $\text{-}18$ to its opposite $18$ to simplify this expression.

$$\begin{array}{c}150-18=132\end{array}$$

Therefore, LaShay's earned a total of $\$132$ for working at the pizza shop last week.

LaShay wants to buy a new dress for her upcoming performance with her earnings for helping her aunt at the pizza shop. She and her parents are at the shopping center. They must take the elevator to the sixth floor to get to LaShay's favorite clothing store.

Solution

LaShay wants to know how many elevator trips it will take before she and her parents can get into the elevator. The elevator can only carry $7$ people per trip. Because there are $35$ people in line ahead of them, dividing this amount by the number of people per trip gives the total number of elevator trips it would take to carry all the people in the line.

$$\begin{array}{c}\text{Number of Elevator Trips}=\frac{35}{7}\end{array}$$

The result is a division of two positive integers. The dividend is $35$ and the divisor is $7.$ Move to the right on a number line, starting from zero and using intervals of size the divisor, until the dividend is reached. The number of intervals will be the quotient.

Since it takes $5$ jumps of $7$ to reach $35,$ $35÷7=5.$ Now, recall that LaShay and her family are at the end of the line. This means that it will take $5$ elevator trips until they get into the elevator. LaShay cannot wait to get her new dress!

LaShay's concert went so well that she decided to play one of her favorite pieces for an upcoming competition.

Solution

Divide the number of points lost by the number of mistakes made to get the number of points LaShay lost for each mistake. Note that she lost $\text{-}96$ points in total and made $6$ mistakes.

$$\begin{array}{c}\text{Points Lost Per Mistake}\\ \text{-}96÷6\end{array}$$

The result is a division of two integers with different sings. Recall that the quotient of two integers with different signs is always negative. With this information in mind, ignore the signs for now and perform the division of whole numbers.

$$\begin{array}{c}\text{-}96÷6\Rightarrow 96÷6\end{array}$$

Because $96$ is a two-digit number, use long division to perform the resulting division.

The quotient is $16.$ However, to get the result of the original division, remember to change $16$ to its opposite, $\text{-}16.$

$$\begin{array}{c}96÷6=16\Rightarrow \text{-}96÷6=\text{-}16\end{array}$$

This means that LaShay lost $16$ points for each mistake made. This did not stop her, though — she took the competition by storm with her incredible talent!