Question

1. solve the shape - equation puzzle.

■+■ = 10
●+★ = 9
●+■ = 8
■ = _ ● = _ ★ = ___

1. find each product in the table.

product
11×1=
11×2=
11×3=
11×4=
11×5=
11×6=
11×7=
11×8=
11×9=
describe a pattern you see in the table. do you think the pattern will continue?

1. some engines require oil to be mixed with gasoline. for every gallon of gas used, 4 ounces of oil should be added. how much gas is needed if 12 ounces of oil are used?
2. sam chose a number. he then multiplied his number by 2 and added 7. his result was 23. what was his original number?
3. kristen bought 2 candy bars for $1.29 each and a soda for $1.99. she had to pay $0.38 in tax. how much was her bill?
4. a building is composed of three rooms as shown. what is the perimeter of the building? 14 ft

4 ft
22 ft
12 ft

Explanation:

1. Solve the shape - equation puzzle

Step1: Find the value of the square

Let the square be \(x\). Given \(x + x=10\), then \(2x = 10\), so \(x=\frac{10}{2}=5\).

Step2: Use the value of the square to find the value of the circle

Let the circle be \(y\). Since \(y + x = 8\) and \(x = 5\), then \(y=8 - 5=3\).

Step3: Use the value of the circle to find the value of the star

Let the star be \(z\). Since \(y+z = 9\) and \(y = 3\), then \(z=9 - 3=6\).

Step1: Calculate the products

\(11\times1 = 11\), \(11\times2=22\), \(11\times3 = 33\), \(11\times4 = 44\), \(11\times5 = 55\), \(11\times6 = 66\), \(11\times7 = 77\), \(11\times8 = 88\), \(11\times9 = 99\).
The pattern is that the product of 11 and a positive - integer \(n\) (where \(n\in\{1,2,\cdots,9\}\)) can be found by writing \(n\) twice. For example, \(11\times2\) gives 22, \(11\times3\) gives 33. This pattern will continue for single - digit non - negative integers because \(11\times n=(10 + 1)\times n=10n + n\).

Step1: Set up a proportion

Let \(g\) be the amount of gas in gallons. We know that the ratio of oil to gas is \(4\) ounces of oil per gallon of gas. If we have 12 ounces of oil, then the proportion is \(\frac{4\text{ ounces of oil}}{1\text{ gallon of gas}}=\frac{12\text{ ounces of oil}}{g\text{ gallons of gas}}\).

Step2: Solve the proportion

Cross - multiply: \(4g=12\), so \(g = 3\) gallons.

Answer:

Square: 5, Circle: 3, Star: 6

2. Find each product in the table