# Which number is increased by 20 corresponds to 144

## Meinstein

Text equations are usually difficult for most students to begin with. But they become easy if you follow certain points - and persist.

1. First of all: read the text carefully!
2. Underline the text that gives exact information (like “triple” or numbers and what they mean).
4. Figure out what to look for.
5. Now write out what is given and what is sought, preferably with the variables.
6. Whenever possible, make a sketch of the situation.
7. Find the formulas and laws that might help you solve the problem.
8. Write the equation.
9. Solve the equation using equivalent transformations.
10. Check the solution with a sample (insert result).
11. Formulate an answer sentence (every word problem requires a text answer!).

### Simple text equations

Three times a number gives exactly 12. What is the name of the number?
Translation of the text into an equation:
3 ⋅ x = 12
Solving the equation
3 ⋅ x = 12 | : 3
x = 4
The number is called 4.

The fivefold of a number increased by 7 results in 32. What is the name of the number?
5 ⋅ x + 7 = 32
Solving the equation
5 ⋅ x + 7 = 32 | -7
5x = 25 | : 5
x = 5
The number is called 5.

Nine times a number less 13 equals 23.
9 ⋅ x - 13 = 23
Solution of the equation:
9 ⋅ x - 13 = 23 | +13
9x = 36 | : 9
x = 4
The number is called 4.

If you decrease a number by 2 and take five times that number, you get 20.
5 ⋅ (x - 2) = 20
Solution of the equation:
5 ⋅ (x - 2) = 2 | encompass.
5x - 10 = 2 | + 10
5x = 12 | : 5
x = 12/5 = 2.4
The number is 2.4.

The double of a number increased by 8 results in the opposite number reduced by 1.
2 ⋅ x + 8 = - x - 1
Solution of the equation:
2 ⋅ x + 8 = - x - 1 | + x
3x + 8 = - 1 | - 8th
3x = - 9 | : 3
x = - 3
The number is called - 3.

The product of a number and 5 is 70.
5x = 70
5x = 70 | : 5
x = 13
The number is 13.

The sum of two consecutive natural numbers is 27.
x + (x + 1) = 27
2x + 1 = 27
2x = 26
x = 13
The number is 13.

The difference between a number and a quarter of the number is 15.
x - x / 4 = 15
3x / 4 = 15
3x = 60
x = 20
The number is called 20.

The product of a number plus 5, increased by 6, is 41.
5x + 6 = 41
5x = 35
x = 7
The number is called 7.

In 17 years Monika will be twice as old as she is now.
x + 17 = 2x I - x
x = 17
She is now 17 years old.

If I triple a number, I get the number increased by 8.
3x = x + 8
2x = 8
x = 4
The number is called 4.

If I multiply a number by itself, I get 121.
x ⋅ x = 121
x = 11
The number is called 11.

If I add the product of 5 and 3 to a number, I get this number around
10 enlarged.
x + 5 ⋅ 3 = x + 10 I - x
15 = 10 is a contradiction.

Susanne and Anna collected 600 nuts together. Anna says: If you give me half the nuts that you have and then I give you a third of the nuts that I have, we will have as many nuts. How many nuts did both have at the beginning?

x + y = 600 nuts
x + 600 - x = 600 nuts

Susanne gives Anna half, Anna gets half.
Susanne Anna
x - x / 2 600 - x + x / 2
x / 2 600 - x / 2
Anna gives Susanne a third back. Then there should be the same number for both.
x / 2 + 200 - x / 6 = 400 - 2x / 6
2x / 6 + 200 = 400 - x / 3
x / 3 + x / 3 = 400-200
2/3 x = 200
2x = 600
x = 300
So at the beginning they both had the same number of nuts, namely 300 each.

Example 2: Numbers of four

Five consecutive numbers of four together give 420. What are their names?

Given
Five numbers
consecutive numbers of four
their sum is 420

x + (x + 4) + (x + 8) + (x + 12) + (x + 16) = 420
5x + 40 = 420
5x = 420 - 40 = 380
x = 76

Sample: 76 + 80 + 84 + 88 + 92 = 420 q.e.d.

The first number is called 76.

Example 3: Mathematical term in words

The difference between the squares of two natural numbers with the difference 3 is 381. What is the name of the smaller of the two numbers?

Difference is one number subtracted from the other
Square means number times number
Natural numbers are the numbers with which we count, i.e. 1, 2, 3, 4 ...
Difference 3 means: there is a difference of 3 between the first and the second number
What is asked is the smaller of the two numbers.

(x + 3)2 - x= 381
x2 + 6x + 9 - x2 = 381
6x + 9 = 381
6x = 372
x = 62

The smaller number is 62.

Example 4: Distance - time - speed task

Two vehicles are approaching each other at speeds of 40 and 60 km / h from two locations that are 50 km apart. The second leaves 30 minutes after the first. Determine when and where they meet.

What I can read from the:
Vehicle A has 40 km / h
Vehicle B has 60km / h
Distance of the descent 50km
B leaves 30 minutes later

Wanted: When and where to meet

I create two equations from this:
Vehicle A: y = 40x
Vehicle B: y = - 60 (x - 0.5) + 50
Equation of the equation:
40x = - 60 (x - 0.5) + 50
40x = - 60x + 30 + 50
100x = 80
x = 80/100 = 0.8
They meet after 0.8 hours or 48 minutes.

Substituting x into the equation for vehicle A:
y = 40 ⋅ 0.8 = 32
They meet 32 ​​kilometers from the departure point of vehicle A (or 18 kilometers from the departure point of vehicle B).

I can also solve the task graphically.
The y-axis then represents the path
the x-axis represents time.
The vehicles describe straight lines in the coordinate system.
Vehicle A is represented by the blue straight line.
Vehicle B through the red straight. It leaves 30 minutes (0.5 hours) later from exactly 50 km away.

### Slightly more difficult text equations

Max and Roger brag about the amount of text messages they send every day. Max sent 25 text messages yesterday. That's 7 SMS more than three times the amount from Roger. How many text messages did Roger send yesterday?

Solution:
Max sent 25 SMS yesterday. Seven more than three times the amount of Roger.
Roger sent x SMS. So 3x + 7 corresponds to Max's number.
25 = 3x + 7
3x = 25 - 7
18 = 3x
x = 6
So Roger sent 6 SMS.
Control:
6 * 3 = 18 Max sends three times more ...
18 + 7 = 25 Max sends 7 less than three times more….

Max pays 15 cents per SMS with his mobile phone subscription. Roger only pays 5 cents per SMS, plus a monthly basic fee of 3 francs for the SMS option. From how many SMS is the SMS option worthwhile for Roger?

Solution:
Max 15 cents per SMS
Roger 5 cents per SMS plus a basic monthly fee of 3 francs.
0.15 * x = 0.05 * x + 3
x is the number of SMS. It is important that the same units are used: Francs.
If the SMS costs are equated in an equation, x means the number of SMS for which the two pay the same amount. We solve for x:
0.15x - 0.05x = 3
0.1x = 3
x = 30
That means, from the 31st SMS, Roger will benefit.
If you have 30 SMS, you both pay the same.
Control:
Max per month: 30 * 0.05 + 3 = 4.50
Roger per month: 30 * 0.15 = 4.50

3. A staircase has 22 steps. If each step were built 1.6 cm higher, two steps could be saved. How high is a level?

Step 1 Read the assignment text carefully!

For a staircase that has to overcome a given height, the following applies: the higher the individual steps, the fewer steps the staircase contains.

Step 2 choosing the unknown
Each step is x cm high.
In other words: x = height of a step in cm

Step 3 Establish the equation
Height of the original stairs = height of the modified stairs (in cm)
We can calculate the height of the stairs in two ways. On the one hand with the 22 existing levels, on the other hand with the 20 higher levels:

Equation: 22 * ​​x = 20 * (x + 1.6cm)

Step 4 Solve the equation
22x = 20x + 32cm
22x - 20x = 32cm
2x = 32cm
x = 16cm
The height of the stair step x is therefore 16cm

Step 5 review the solution
We do the math: 22 steps of 16 cm result in a height of 22 * ​​16 cm = 352 cm
20 steps of (16 + 1.6) cm result in a height of 20 * 17.6 cm = 352 cm

Step 6 text response
Each step is 16 cm high.