Replacing an Operation with Other Operations

This chapter covers:

techniques for mental multiplication and division
approximations of the number π

0.5 and 2

Multiplying by 0.5 and dividing by 2 give the same result.

Dividing by 0.5 and multiplying by 2 give the same result.

Notice

n is used as a placeholder for a generalised example. When solving specific equations, it is substituted with the necessary number.

$n : 2 = \frac{n}{2} = \frac{1}{2} \cdot n$
$n : 0.5 = n : \frac{1}{2} = n \cdot \frac{2}{1} =$
$= \frac{2 n}{1} = 2 n$

Example 1

52 : 0.5 = 52 · 2 = 104
43 · 0.5 = 43 : 2 = 21.5

Drag the operations to their corresponding columns.

0.5 · 0.5
32 : 0.5
0.5 : 0.5
0.5 · 23
12 : 2
17 · 2
65 : 0.5
18 · 0.5

The answers to the operations are given below. Select the numbers among them that are not the answers to the given operations.

8.5
11.5
64
6
1
32.5
130
0.25
46
9
25
34

2 : 0.5
0.2 · 0.5
2 · 0.5
1 : 0.5
0.2 : 0.5
0.5 · 0.5

15 : 0.5
150 · 0.5
150 : 0.5
15 · 0.5
1.5 : 0.5

Seotud sisu

Multiplying by 1.5 and 2.5

Multiplying by 3 and then dividing by 2 gives the same result as multiplying by 1.5.

Multiplying by 5 and then dividing by 2 gives the same result as multiplying by 2.5.

Notice

n is a placeholder for a freely chosen number.

$1.5 \cdot n = 1 \frac{1}{2} \cdot n =$
$= \frac{3}{2} \cdot n = \frac{3 n}{2} = \frac{n}{2} \cdot 3 =$
$= n : 2 \cdot 3$
$2.5 \cdot n = 2 \frac{1}{2} \cdot n =$
$= \frac{5}{2} \cdot n = \frac{5 n}{2} = \frac{n}{2} \cdot 5 =$
$= n : 2 \cdot 5$

Example 2

If I need to multiply 14 by 1.5, I first divide 14 by 2 to equal 7, and then multiply 7 by 3 to equal 21, so I get 1.5 · 14 = 21.
If I need to multiply 72 by 2.5, I first divide 72 by 2 to equal 36, and then multiply 36 by 5 to equal 180, so I get 2.5 · 72 = 180.

1.5 · 28 = =
1.5 · 46 = =
1.5 · 14 = =
1.5 · 17 = =
1.5 · 61 = =
2.5 · 8 = =
2.5 · 22 = =
2.5 · 34 = =
2.5 · 52 = =
2.5 · 9 = =

2.5 · 12 = 12 : 2 · =
1.5 · 12 = 12 : 2 · =
3.5 · 12 = 12 : 2 · =
4.5 · 12 = 12 : 2 · =
5.5 · 12 = 12 : 2 · =
3.5 · 26 = 26 : 2 · =
4.5 · 13 = 13 : 2 · =
5.5 · 84 = 84 : 2 · =

Notice

$⍰ .5 \cdot n = \frac{2 \cdot ⍰ + 1}{2} \cdot n = = \frac{n}{2} \cdot (2 \cdot ⍰ + 1) = = n : 2 \cdot (2 \cdot ⍰ + 1)$

Seotud sisu

Divisiding by 25

Multiplying by 4 and then dividing by 100 gives the same result as dividing by 25.

Notice

n is a freely chosen number.

$n : 25 = n : \frac{100}{4} = n \cdot \frac{4}{100}$

300 : 25 =

850 : 25 =

625 : 25 =

175 : 25 =

105 : 25 =

70 : 25 =

90 : 25 =

275 : 25 =

Seotud sisu

Multiplying by 11

When multiplying a two-digit number by 11, we know that 11 = 10 + 1, so 11 · n = (10 + 1) · n = 10 · n + n.

Example 3

11 · 34 and 11 · 97

Replacing the multiplication of 11 · 34 and 11 · 97 with addition when doing a mental calculation.

11 · 33 =

11 · 35 =

11 · 62 =

11 · 75 =

11 · 98 =

11 · 78 =

22 · 31 =

22 · 45 =

22 · 23 =

22 · 14 =

22 · 71 =

22 · 88 =

Seotud sisu

Practice and solve

We know that the number π is the quotient of the length of a circle and its diameter. The closest fraction to this value is
π ≈ $\frac{22}{7}$ ≈ 3.1428571... .

If d = 3, then
C ≈ $\frac{}{7}$ ≈ .
If d = 12, then
C ≈ $\frac{}{7}$ ≈ .
If d = 50, then
C ≈ $\frac{}{7}$ ≈ .
If d = 28, then
C ≈ $\frac{}{7}$ ≈ .
If d = 41, then
C ≈ $\frac{}{7}$ ≈ .
If d = 39, then
C ≈ $\frac{}{7}$ ≈ .

Find the circumference of 50 cents.

How can you make sure? You can, for example, roll the coin over a ruler.

π ≈ $\frac{22}{7}$

Trees are companions to humanity. Their wood, roots and leaves are used around the world. As long as we plant seeds and care for them, and not take trees that are essential to their ecosystem (such as those offering shelter to animals), we can continue to have a respectful relationship with trees. One example of this is waiting for a tree to be large enough to yield much wood, and old enough to have contributed much to their ecosystem.

A mature pine has a diameter of 28 cm and a circumference of cm.
A mature black poplar has a diameter of 182 cm and a circumference of cm.

A pine tree is ready to harvest when their diameter is the longer side of a sheet of paper. Their girth is then around cm (rounded up to a whole number).

The oak tree is ready to harvest when they are
Birch is ready to harvest when they are
The fir tree ready to harvest when they are
The alder is ready to harvest when they are
The pine tree is ready to harvest when they are
The hanging alder is ready to harvest when they are

Seotud sisu

Rules and formulas

$n \cdot 0.5 = \frac{n}{2} = n : 2$
$n : 0.5 = 2 n$
$1.5 n = \frac{3 n}{2} = n : 2 \cdot 3$
$n : 25 = n : \frac{100}{4} = n \cdot \frac{4}{100} =$ $n : 100 \cdot 4$
$11 n = (10 + 1) n = 10 n + n$

Seotud sisu

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Replacing an Operation with Other Operations

This chapter covers:

Seotud sisu

0.5 and 2

Notice

Example 1

Seotud sisu

Multiplying by 1.5 and 2.5

Notice

Example 2

Notice

Seotud sisu

Divisiding by 25

Notice

Seotud sisu

Multiplying by 11

Example 3

Seotud sisu

Practice and solve

Seotud sisu

Rules and formulas

Seotud sisu

Lisa materjali

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