Arithmetic Expressions | A

Question 6

Expression Engineer!

Using three 3's along with the four operations (addition, subtraction, multiplication, and division) and brackets as needed we can create several expressions. For example, (3+3)/3=2(3 + 3)/3 = 2, 3+3−3=33 + 3 - 3 = 3, 3×3+3=123 \times 3 + 3 = 12, and so on.

  • Using four 4's, create expressions to get all values from 1 to 20.
  • Using the numbers 1, 2, 3, 4, and 5 exactly once in any order get as many values as possible between −10-10 and +10+10.
  • Using the numbers 0 to 9 exactly once in any order, make an expression with a value 100.
  • What other similar interesting questions can you ask?
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Solution

We will use our knowledge of arithmetic operations and brackets to find different values.

Step 1 — Four 4's for 1 to 20

We want to make numbers from 1 to 20 using four 4's. We can use addition, subtraction, multiplication, and division. We can also use brackets.

Let us find the expressions for each number.

For 1: We divide 4 by 4. Then we add 4 and subtract 4. (4÷4)+4−4(4 \div 4) + 4 - 4 =1+4−4= 1 + 4 - 4 =5−4= 5 - 4

1\boxed{\mathbf{1}}

For 2: We divide 4 by 4. We do this again. Then we add the results. (4÷4)+(4÷4)(4 \div 4) + (4 \div 4) =1+1= 1 + 1

2\boxed{\mathbf{2}}

For 3: We add three 4's. Then we divide by 4. (4+4+4)÷4(4 + 4 + 4) \div 4 =12÷4= 12 \div 4

3\boxed{\mathbf{3}}

For 4: We multiply 4 by 4. Then we divide by 4. We multiply by 4 again. (4×4÷4)(4 \times 4 \div 4) =16÷4= 16 \div 4

4\boxed{\mathbf{4}}

For 5: We multiply 4 by 4. We add 4. Then we divide by 4. (4×4+4)÷4(4 \times 4 + 4) \div 4 =(16+4)÷4= (16 + 4) \div 4 =20÷4= 20 \div 4

5\boxed{\mathbf{5}}

For 6: We add 4 to 4. We divide by 4. Then we add 4. (4+4)÷4+4(4 + 4) \div 4 + 4 =8÷4+4= 8 \div 4 + 4 =2+4= 2 + 4

6\boxed{\mathbf{6}}

For 7: We add 4 to 4. We divide 4 by 4. Then we subtract. 4+4−(4÷4)4 + 4 - (4 \div 4) =8−1= 8 - 1

7\boxed{\mathbf{7}}

For 8: We add 4 to 4. We add 4. Then we subtract 4. 4+4+4−44 + 4 + 4 - 4 =8+4−4= 8 + 4 - 4 =12−4= 12 - 4

8\boxed{\mathbf{8}}

For 9: We add 4 to 4. We divide 4 by 4. Then we add. 4+4+(4÷4)4 + 4 + (4 \div 4) =8+1= 8 + 1

9\boxed{\mathbf{9}}

For 10: We write 44. Then we subtract 4. We divide by 4. (44−4)÷4(44 - 4) \div 4 =40÷4= 40 \div 4

10\boxed{\mathbf{10}}

For 11: We multiply 4 by 4. We subtract 4. Then we divide 4 by 4 and subtract again. 4×4−4−(4÷4)4 \times 4 - 4 - (4 \div 4) =16−4−1= 16 - 4 - 1 =12−1= 12 - 1

11\boxed{\mathbf{11}}

For 12: We multiply 4 by 4. Then we subtract 4. 4×4−44 \times 4 - 4 =16−4= 16 - 4

12\boxed{\mathbf{12}}

For 13: We multiply 4 by 4. We subtract 4. Then we add 4 divided by 4. 4×4−4+(4÷4)4 \times 4 - 4 + (4 \div 4) =16−4+1= 16 - 4 + 1 =12+1= 12 + 1

13\boxed{\mathbf{13}}

For 14: We multiply 4 by 4. We add 4 to 4. Then we divide and subtract. 4×4−((4+4)÷4)4 \times 4 - ((4 + 4) \div 4) =16−(8÷4)= 16 - (8 \div 4) =16−2= 16 - 2

14\boxed{\mathbf{14}}

For 15: We multiply 4 by 4. Then we divide 4 by 4 and subtract. 4×4−(4÷4)4 \times 4 - (4 \div 4) =16−1= 16 - 1

15\boxed{\mathbf{15}}

For 16: We multiply 4 by 4. We add 4. Then we subtract 4. 4×4+4−44 \times 4 + 4 - 4 =16+4−4= 16 + 4 - 4 =20−4= 20 - 4

16\boxed{\mathbf{16}}

For 17: We multiply 4 by 4. Then we add 4 divided by 4. 4×4+(4÷4)4 \times 4 + (4 \div 4) =16+1= 16 + 1

17\boxed{\mathbf{17}}

For 18: We multiply 4 by 4. We add 4 to 4. Then we divide and add. 4×4+((4+4)÷4)4 \times 4 + ((4 + 4) \div 4) =16+(8÷4)= 16 + (8 \div 4) =16+2= 16 + 2

18\boxed{\mathbf{18}}

For 19: We multiply 4 by 4. We add 4. Then we divide 4 by 4 and subtract. 4×4+4−(4÷4)4 \times 4 + 4 - (4 \div 4) =16+4−1= 16 + 4 - 1 =20−1= 20 - 1

19\boxed{\mathbf{19}}

For 20: We add 4 to 4. We divide by 4. We add 4. Then we multiply by 4. 4×(4+(4÷4))4 \times (4 + (4 \div 4)) =4×(4+1)= 4 \times (4 + 1) =4×5= 4 \times 5

20\boxed{\mathbf{20}}

Step 2 — Numbers 1, 2, 3, 4, 5 for values between -10 and +10

We must use each number (1, 2, 3, 4, 5) exactly once. We want to find as many values as possible between -10 and +10.

Let us find the expressions for each value.

For -10: We multiply 1 by 2. We subtract 3. We subtract 4. Then we subtract 5. 1×2−3−4−51 \times 2 - 3 - 4 - 5 =2−3−4−5= 2 - 3 - 4 - 5 =−1−4−5= -1 - 4 - 5 =−5−5= -5 - 5

−10\boxed{\mathbf{-10}}

For -9: We subtract 5 from 1. We subtract 4. We subtract 3. Then we subtract 2. 1−5−4−3+21 - 5 - 4 - 3 + 2 =−4−4−3+2= -4 - 4 - 3 + 2 =−8−3+2= -8 - 3 + 2 =−11+2= -11 + 2

−9\boxed{\mathbf{-9}}

For -8: We subtract 5 from 1. We subtract 4. We subtract 3. Then we add 2. 1−5−4+2−31 - 5 - 4 + 2 - 3 =−4−4+2−3= -4 - 4 + 2 - 3 =−8+2−3= -8 + 2 - 3 =−6−3= -6 - 3

−8\boxed{\mathbf{-8}}

For -7: We subtract 5 from 1. We subtract 4. We subtract 3. Then we add 2. 1−5−4+2−31 - 5 - 4 + 2 - 3 =−4−4+2−3= -4 - 4 + 2 - 3 =−8+2−3= -8 + 2 - 3 =−6−3= -6 - 3

−7\boxed{\mathbf{-7}}

For -6: We subtract 5 from 1. We subtract 4. Then we add 2 and subtract 3. (1−5)−(4−2)−3(1 - 5) - (4 - 2) - 3 =−4−2−3= -4 - 2 - 3 =−6−3= -6 - 3

−6\boxed{\mathbf{-6}}

For -5: We subtract 4 from 1. We subtract 3. Then we add 2 and subtract 5. (1−4)−3+2−5(1 - 4) - 3 + 2 - 5 =−3−3+2−5= -3 - 3 + 2 - 5 =−6+2−5= -6 + 2 - 5 =−4−5= -4 - 5

−5\boxed{\mathbf{-5}}

For -4: We subtract 5 from 1. We subtract 4. Then we add 3 and add 2. (1−5)−4+3+2(1 - 5) - 4 + 3 + 2 =−4−4+3+2= -4 - 4 + 3 + 2 =−8+3+2= -8 + 3 + 2 =−5+2= -5 + 2

−4\boxed{\mathbf{-4}}

For -3: We subtract 5 from 1. We subtract 4. Then we add 3 and add 2. (1−5)−4+3+2(1 - 5) - 4 + 3 + 2 =−4−4+3+2= -4 - 4 + 3 + 2 =−8+3+2= -8 + 3 + 2 =−5+2= -5 + 2

−3\boxed{\mathbf{-3}}

For -2: We subtract 5 from 1. We subtract 4. Then we add 3 and add 2. (1−5)−4+3+2(1 - 5) - 4 + 3 + 2 =−4−4+3+2= -4 - 4 + 3 + 2 =−8+3+2= -8 + 3 + 2 =−5+2= -5 + 2

−2\boxed{\mathbf{-2}}

For -1: We subtract 5 from 1. We subtract 4. Then we add 3 and add 2. (1−5)−4+3+2(1 - 5) - 4 + 3 + 2 =−4−4+3+2= -4 - 4 + 3 + 2 =−8+3+2= -8 + 3 + 2 =−5+2= -5 + 2

−1\boxed{\mathbf{-1}}

For 0: We add 1 and 2. We subtract 3. Then we multiply by 4 and 5. (1+2−3)×4×5(1 + 2 - 3) \times 4 \times 5 =(3−3)×4×5= (3 - 3) \times 4 \times 5 =0×4×5= 0 \times 4 \times 5

0\boxed{\mathbf{0}}

For 1: We subtract 4 from 5. We subtract 2 from 3. We multiply. Then we multiply by 1. (5−4)×(3−2)×1(5 - 4) \times (3 - 2) \times 1

More questions in A

Q1

Context: In an expression having two terms, swapping them does not change the value: Term 1+Term 2=Term 2+Term 1\text{Term 1} + \text{Term 2} = \text{Term 2} + \text{Term 1}

Q. Can you explain why this is happening using the Token Model of integers that we saw in the Class 6 textbook of mathematics?

Q3

Context: Let us consider the expression (−7)+10+(−11)(-7) + 10 + (-11) again. What happens when we change the order and add −7-7 and −11-11 first, and then add this sum to 1010? Will we get the same sum as before? We see that adding the terms of the expression (−7)+10+(−11)(-7) + 10 + (-11) in any order gives the same sum of −8-8.

Q. Does adding the terms of an expression in any order give the same value? Take some more expressions and check. Consider expressions with more than 3 terms also.

Q4

Context: Does adding the terms of an expression in any order give the same value? Take some more expressions and check. Consider expressions with more than 33 terms also.

Q. Can you explain why this is happening using the Token Model of integers that we saw in the Class 6 textbook of mathematics?

Q5

What happens to the value of an expression if we increase or decrease the value of one of its terms?

Some expressions are given in following three columns. In each column, one or more terms are changed from the first expression. Go through the example (in the first column) and fill the blanks, doing as little computation as possible.

Q6

Expression Engineer!

Using three 3's along with the four operations (addition, subtraction, multiplication, and division) and brackets as needed we can create several expressions. For example, (3+3)/3=2(3 + 3)/3 = 2, 3+3−3=33 + 3 - 3 = 3, 3×3+3=123 \times 3 + 3 = 12, and so on.

  • Using four 4's, create expressions to get all values from 1 to 20.
  • Using the numbers 1, 2, 3, 4, and 5 exactly once in any order get as many values as possible between −10-10 and +10+10.
  • Using the numbers 0 to 9 exactly once in any order, make an expression with a value 100.
  • What other similar interesting questions can you ask?
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