Terry's Tire Store sells automobile and truck tires through...

Terry's Tire Store sells automobile and truck tires through three retail outlets. Sales at the Cahokia store for the months of January, February, and March broke down as follows: $350,420,$ and 530 auto tires and $220,180,$ and 140 truck tires. The Shady Oak branch sold $430,560,$ and 690 auto tires and 280 $320,$ and 220 truck tires during the same 3 months. Sales figures for the downtown store were 864,980 , and 1236 auto tires and $535,542,$ and 332 truck tires. a. Write a $2 \times 3$ "sales matrix" for each store $[C \rightarrow \text { Cahokia, } s \rightarrow \text { Shady } \text { Oak, } D \rightarrow$ Downtown], with January, February, and March columns, and two rows showing the sales of auto and truck tires respectively. b. Use the matrices from Part (a) to determine how many more or fewer tires of each type the downtown store sold (each month) over the other two stores combined. c. Market trends indicate that for the same three months in the following year, the Cahokia store will likely experience a $10 \%$ increase in sales, the Shady Oak store a $3 \%$ decrease, with sales at the downtown store remaining level (no change). What will be the combined monthly sales from all three stores next year, for each type of tire?

Terry's Tire Store sells automobile and truck tires through three retail outlets. Sales at the Cahokia store for the months of January, February, and March broke down as follows: $350,420,$ and 530 auto tires and $220,180,$ and 140 truck tires. The Shady Oak branch sold $430,560,$ and 690 auto tires and 280 $320,$ and 220 truck tires during the same 3 months. Sales figures for the downtown store were 864,980 , and 1236 auto tires and $535,542,$ and 332 truck tires.
a. Write a $2 \times 3$ "sales matrix" for each store $[C \rightarrow \text { Cahokia, } s \rightarrow \text { Shady } \text { Oak, } D \rightarrow$ Downtown], with January, February, and March columns, and two rows showing the sales of auto and truck tires respectively.
b. Use the matrices from Part (a) to determine how many more or fewer tires of each type the downtown store sold (each month) over the other two stores combined.
c. Market trends indicate that for the same three months in the following year, the Cahokia store will likely experience a $10 \%$ increase in sales, the Shady Oak store a $3 \%$ decrease, with sales at the downtown store remaining level (no change). What will be the combined monthly sales from all three stores next year, for each type of tire?

Key Concepts

Matrix Representation

Matrices are organized arrays of numbers arranged into rows and columns. They provide a systematic way to structure and present data, which is particularly useful in applications like organizing sales figures or other multidimensional data sets.

Matrix Operations

Matrix operations such as addition, subtraction, and element-wise manipulation allow for the combination and comparison of datasets. These operations help in analyzing differences or summing values across corresponding entries in different matrices.

Scalar Multiplication

Scalar multiplication involves multiplying every element of a matrix by a constant factor. This is useful for modeling percentage increases or decreases in data, such as adjusting previous sales records to predict future figures.

Data Aggregation and Comparison

Aggregating data involves combining multiple matrices to get a composite view of the data, often through addition. Comparison is achieved by subtracting one matrix from another to determine relative performance differences between datasets.

Transcript

00:01 All right, so this question gives us the data for the sales at a tire store, given there are three locations, and the number of auto tires and truck tires that they sold in january, february, and march.

00:14 In part a of this question, we're asked to create a two -by -three sales matrix for each store.

00:21 So we have store c, store s, and store d, given the number of auto and truck tires they sold in january, february, in march.

00:33 So we are told that for store c, 350 auto tires are sold in january, 420 auto tires are sold in february, and 530 auto tires are sold in march.

00:51 So you can see i filled this in in the first row of our 2x3 matrix for auto tires.

00:55 And then each column is the number of tires that was sold in the corresponding month.

01:00 For truck tires at this store, we sold 220 in january, 180 in february, and 140 in march.

01:10 We're going to go ahead and repeat this process for store s.

01:14 So we are told in the question that 430 auto tires are sold in january.

01:19 So i felt that in row one, column one corresponding with auto and january.

01:25 560 auto tires were sold in february, and 690 auto tires were sold in march.

01:30 For truck tires, 280 were sold in january, 320 were sold in february, and 220 were sold in march.

01:42 Finally, for our last store, store d, 864 tires were sold in january for auto, 980 auto tires were sold in february, and 1236 were sold in march.

02:01 For truck tires, 535 are sold in january, 542 were sold in february, and 332 were sold in merch.

02:11 So this completes part a, where we had to create three 2x3 matrices using the given data for each store.

02:20 Now moving over to part b, we are asked the difference between the number of tires that the downtown store sold.

02:32 Store d versus the other two stores combined.

02:36 So to do so, i'm going to subtract the combined sales from stores c and s from store d.

02:46 So to tell the difference between d and the other two stores combined.

02:50 Again, we're going to create a two by three matrix to display this result.

02:56 But before i do this, i'm first just going to calculate the combined stores.

03:00 So matrix c plus s.

03:03 To do so, i just add corresponding components.

03:06 So row 1, column 1 of c plus row 2, row 1, column 1 of s is 350 plus 430, which gives us 780.

03:17 Column 2, 420 plus 560 gives me 980.

03:22 Column 3, 530 plus 690 gives me 1220.

03:27 Moving into the second row, 220 plus 280 is 500.

03:31 180 plus 320 is also 500, and 140 plus 220 is 360.

03:39 Now i'm going to find d minus this combined matrix.

03:44 So to find this matrix, we subtract corresponding components in c plus s, our combined matrix, from d.

03:55 In row 1, column 1, we would have 864 minus 780, which gives me 84, column 2, 90d minus 90d is 0.

04:07 Column 3, 1236 minus 1220 gives me 16...

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