# Numerical Expressions and Order of Operations.

Numerical expression is a collection of numerals with one or more operation symbols.

 Example: 3 x 5; 7 + 8; 35 ÷ 7 + 10; (65 - 45 ÷ 9) ÷ 4

If you do all operations in the numerical expression, you end up with a single numeral that is called the value of numerical expression. The value of all the numerical expressions above is 15.

You should do arithmetic operations in particular order.

The rules are:

1. If the expression contains brackets, first calculate whatever is within the brackets and then work outward.
2. Working from left to right, do multiplication and division before addition and subtraction.

Compare:

 3 1 4 2 12 - 8 ÷ 4 + 3 x 2 = 12 - 2 + 6 = 16 1 2 4 3 (12 - 8) ÷ 4 + 3 x 2 = 4 ÷ 4 + 6 = 1 + 6 = 7 4 1 2 3 12 - (8 ÷ 4 + 3) x 2 = 12 - (2 + 3)x 2 = 12 - 5 x 2 = 12 - 10 = 2

You should not try to find the answer in your head right away when you calculate such long numerical expressions. Notice how they are solved above, in steps. Write a chain of numerical expressions that are equal to each other.

Notice that rule 2. does not mean that you do multiplication before division or addition before subtraction.
Multiplication and division have the same level of priority, and you should do them in order, working from left to right:

 1 2 2 1 Compare: 12 ÷ 2 x 6 = 6 x 6 = 36 and 12 ÷ (2 x 6) = 12 ÷12=1

If you do multiplication first in the first example, the result will be 1 instead of 36.

In the same way, addition and subtraction have the same level of priority, and you should do them in order, working from left to right:

 1 2 2 1 Compare: 12 - 2 + 6 = 10 + 6 = 16 and 12 - (2 + 6) = 12 - 8 = 4

If you do addition first in the first example, the result will be 4 instead of 16.

### Exercises.

1. Fill in the missing numbers:

a) 64 - 7 x 2 + 8 ÷4 - 1 = 64 - + - 1 =

b) (24 - 8 ÷ 2) x 2 + 6 ÷ 3 = (24 - ) x 2 + 6 ÷3 =
x 2 + 6
÷ 3 = + =

c) 44 - (7 x 2 + 15÷ 5 - 2 ) = 44 - ( + - 2) =
44 - =

d) 22 + 48 ÷ 6 + 18 ÷ (9 - 3) = 22+ 48 ÷ 6 + 18 ÷ =
22 + + =

e) (60 - 44) x (12 - 32 ÷ 8) = x (12- ) =
x =

f) 42 + (28 ÷ 7 + 3x 2) ÷ 5 = 42 + ( + ) ÷ 5 = 42 + ÷ 5 =
42 + =

2. Find values of numerical expressions:

 a) 54 + 18 ÷ 3 - 2 = 54 x 18 ÷ 3 x 2 = 54 ÷ 18 + 3 x 2 = 54 ÷ 18 x (3 + 2) = 54 ÷ 18 x 3 - 2 = 54 ÷ (18 ÷ 3) x 2) = b) 36 - 12 ÷ 3 + 3 = (36 - 12) ÷ 3 + 3 = 36 x 12 ÷ 3 x 3 = 36 x 12 ÷ (3 x 3) = 36 ÷ 12 ÷ 3 x 3 = 36 ÷ (12 ÷ 3) x 3 =

3. Put signs +, -, x, ÷ and brackets in the expressions so that the expressions become correct.

 1 2 3 4 = 1 4 3 2 1 = 6 1 2 3 4 = 2 4 3 2 1 = 7 1 2 3 4 = 3 4 3 2 1 = 8 1 2 3 4 = 4 4 3 2 1 = 9 1 2 3 4 = 5 4 3 2 1 = 10

4. Using the signs of addition, subtraction, multiplication and division, and the numbers 2, 3, 5 and 6 in any order, get the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9 and 10 in as many ways as you can. You must use all 4 numbers each time!

Example: (3 - 2) x (6 - 5) = 1; 5 x 2 - 6 - 3 = 1; etc.

5. Put signs +, -, x, ÷ and brackets in the equalities so that they become correct.

 1 2 3 4 5 = 100 5 5 5 5 5 = 8 5 5 5 5 5 = 20

6. Put brackets in the equalities, so that they become correct. Show why you made this choice.

 7 x 9 + 12 ÷ 3 - 2 = 23 7 x 9 + 12 ÷ 3 - 2 = 75

7. Put signs +, -, x, ÷ in the equalities so that they become correct.

 23 19 = 42 56 18 = 38 48 27 = 21 16 18 = 34 6 4 3 = 27 5 6 3 = 27 (8 3) 9 = 45 5 2 9 = 19

8.
It is possible to make 100 with 1 and five 5. Place arithmetic signs and brackets in the equality to make it true.

 1 5 5 5 5 5 = 100

### Visual thinking.

1) In Grid A below, the circles have been placed so that no circle is in the same row or in the same column as any other circle. However, two circles are in the same diagonal. Place four circles on Grid B so that no circles are in the same row, column or diagonal.

2) Show how to cut up a 5x5 square along the grid lines into 7 non-congruent rectangles. You can count squares as rectangles. ( non-congruent means of different sizes. For example, rectangles #1 and #2 are non-congruent, but rectangles #2 and # 3 are congruent)

Hint: First figure out all rectangles that you can use and find their areas. Then chose those 7 rectangles whose total area is 25 squares.

3) The cells of the square have been painted over as shown in the picture. How many more painted cells are there than unpainted cells?

## TEST

Question 1:

 Arrange the digits 1,6,4,3,9,2,1 into the greatest possible seven-digit number. You should use each digit only once.

Question 2:

 It is possible to make 100 with 1 and five same digits other than 5(recall problem 3.8 from the lecture).1 = 100 You can use the brackets and any arithmetic signs. What is this digit? Try to find more than one answer. Show how you got the solution. Use x for multiplication and / for division.

Question 3:

 Place brackets in the following expression so that its value is the greatest possible: 33+33+33+33+3. What is this value?

Question 4:

 Mark was supposed to add 3 to some number and then multiply the result by 6. Instead, he added 6 to the number and then multiplied the result by 3. His answer was 57. What answer should Mark get if he does the problem correctly?

Question 5:

 A goldsmith has a little golden cube whose sides measure 4 cm. He slices this cube into little cubes whose sides measure 1cm. How many little cubes does he get?

Question 6:

 The cells of the 99 x 99 square have been painted over as shown in the picture. How many more painted cells are there than unpainted cells?