Summation

### 1.1. Knowledge to remember

Addition of natural numbers, fractions, and decimals all have the following properties:

Commutative property: a + b = b + a

Associative property: (a + b) + c = a + (b + c)

Plus 0: a + 0 = 0 + a = a

### 1.2. Solve textbook exercises on pages 158, 159

**Lesson 1 Textbook page 158**

Calculate

a) 889972 + 963088

b) \(\frac{5}{6} + \frac{7}{{12}}\)

c) \(3 + \frac{5}{7}\)

d) 926.83 + 549.67

__Solution guide:__

a) 889972 + 96308 = 986280

b) \(\frac{5}{6} + \frac{7}{{12}} = \frac{{10}}{{12}} + \frac{7}{{12}} = \frac {{17}}{{12}}\)

c) \(3 + \frac{5}{7} = \frac{{21}}{7} + \frac{5}{7} = \frac{{26}}{7}\)

d) 26.83+549.67=1476.5

Lesson 2 Textbook page 158

Calculate by the most convenient way:

a) (689 + 875) + 125 581 + (878 + 419)

b) \(\left( {\frac{2}{7} + \frac{4}{9}} \right) + \frac{5}{7};\frac{{17}}{{11} } + \left( {\frac{7}{{15}} + \frac{5}{{11}}} \right)\)

c) 5.87 + 28.69 + 4.13 ; 83.75 + 46.98 + 6.25

__Solution guide:__

a)

\(\begin{array}{l}

(689 + 875) + 125 = 689 + (875 + 125)\\

= 689 + 1000 = 1689\\

581 + (878 + 419) = (581 + 419) + 878\\

= 1000 + 878 = 1878

\end{array}\)

b)

\(\begin{array}{l}

\left( {\frac{2}{7} + \frac{4}{9}} \right) + \frac{5}{7} = \left( {\frac{2}{7} + \frac {5}{7}} \right) + \frac{4}{9} = \frac{7}{7} + \frac{4}{9} = 1 + \frac{4}{9} = 1 \frac{4}{9}\\

\frac{{17}}{{11}} + \left( {\frac{7}{{15}} + \frac{5}{{11}}} \right) = \left( {\frac{ {17}}{{11}} + \frac{5}{{11}}} \right) + \frac{7}{{15}}{\mkern 1mu} {\kern 1pt} = \frac{{ 22}}{{11}} + \frac{7}{{15}} = 2 + \frac{7}{{15}} = 2\frac{7}{{15}}

\end{array}\)

c)

\(\begin{array}{l}

5.87 + 28.69 + 4.13 = (5.87 + 4.13) + 28.69\\

= 10 + 28.69 = 38.69\\

83.75 + 46.98 + 6.25 = (83.75 + 6.25) + 46.98\\

= 90 + 46.98 = 136.98

\end{array}\)

Lesson 3 Textbook page 159

Without performing the calculation, state the prediction of finding x:

a) x + 9.68 = 9.68

b) \(\frac{2}{5} + x = \frac{4}{{10}}\)

__Solution guide:__

a) x + 9.68 = 9.68

Derive x = 0, because 0 plus any number equals the same number.

b) \(\frac{2}{5} + x = \frac{4}{{10}}\)

We have: \(\frac{4}{{10}} = \frac{{4:2}}{{10:2}} = \frac{2}{5}\)

From there we have: \(\frac{2}{5} + x = \frac{2}{5}\)

Derive x = 0, because 0 plus any number equals the same number.

Lesson 4 Textbook page 159

The first faucet flows every hour \(\frac{1}{5}\) the volume of the tank, the second faucet every hour flows \(\frac{3}{10}\) the volume of the tank. What is the volume of the tank when both faucets flow into the tank for one hour?

__Solution guide:__

The number of volume fractions of the tank that can be drained by both faucets per hour is:

\(\frac{1}{5} + \frac{3}{{10}} = \frac{5}{{10}}\) (volume of the tank)

\(\frac{5}{{10}} = 0.5 = 50{\rm{\% }}\)

Answer: 50% of tank volume.

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