A kosarad üres
Solving problems with equations I
In reality, the questions are not usually raised in the form of equations, we only apply them as tools to give answers.
Example 1
We managed to store \latex{ 1,000 } volumes of data on four CD disks. If we stored \latex{ 15 } more on the first, \latex{ 70 } fewer on the second, twice as many on the third, and half as many on the fourth disk, then there would be the same amount of volumes on each disk. How many volumes are stored on each disk separately?
Solution
Let us choose the equal values after changing the volume numbers as the unknown, and let it denote by \latex{ x }. The volume numbers found on the CDs can be expressed with the help of \latex{ x }.
By knowing the total number of volumes we can set up the following equation:
By knowing the total number of volumes we can set up the following equation:
\latex{\begin{align*}(x-15)+(x+70)+\frac{x}{2}+2x=1,000,\\ \frac{9}{2}x+55=1,000,\\ \frac{9}{2}x=945,\\ x=210.\end{align*}}
Thus the equal number of volumes after the change is 210 volumes, therefore the number of volumes found on each CD is \latex{ 195, \, 280,\, 105 } and \latex{ 420 } volumes respectively.
Checking: \latex{195 + 280 + 105 + 420 = 1,000}.
CD
Number of volumes
1
2
3
4
\latex{x-15}
\latex{x+70}
\latex{\frac{x}{2}}
\latex{2x}
Example 2
A father is twice as old as his son. Ten years ago he was three times as old as his son. How old is the father and the son now?
Solution
If the son is \latex{ x } years old now, then the father is \latex{ 2x } years old. Ten years ago the son was \latex{x - 10}, and the father was \latex{2x - 10} years old. At this time the father was three times as old as his son, therefore: \latex{3(x - 10) = 2x – 10}.
The solution of the equation:
\latex{3x-30=2x-10},
\latex{x=20}.
Therefore the son is \latex{ 20 } years old now, whereas his father is \latex{ 40 } years old. \latex{ 10 } years ago their ages were \latex{ 10 } and \latex{ 30 } years respectively, i.e. the father was indeed three times as old as his son.
Father
Son
Now
10 years
ago
ago
\latex{2x}
\latex{x}
\latex{2x-10}
\latex{x-10}
Example 3
We could cover our expenses at a trip as follows. We spent \latex{\frac{1}{3}} of our money and another 9 EUR on the first day, we spent \latex{\frac{1}{3}} of the remaining amount and another 6 EUR on the second day, so we were left with 14 EUR on the last, third day. How much money did we take with us to the trip?
Solution
We can find the missing amount by going “back” in time and by giving the amount we had the previous day with elimination.
Let \latex{ x } denote the money taken on the trip.
- We had \latex{ 14 } EUR on the third day.
- Since \latex{14 + 6 = 20} EUR was the \latex{\frac{2}{3}} of money we had, therefore we had \latex{(14+6)\times\frac{3}{2}=30} at the beginning of the second day.
- On the first day we had \latex{(30+9)\times \frac{3}{2}=58.50} EUR with us for similar reasons.
Let \latex{ x } denote the money taken on the trip.
the solution of the exercise with elimination
Money spent
Remaining amount of money
Day 1
Day 2
Day 3
\latex{\frac{1}{3}x+9}
\latex{x-\left(\frac{1}{3}x+9\right)=\frac{2}{3}x-9}
\latex{\frac{1}{3}\times \left(\frac{2}{3}x-9\right)+6=\frac{2}{9}x+3}
\latex{\frac{2}{3}x-9-\left(\frac{2}{9}x-3\right)=\frac{4}{9}x-12}
\latex{14}
The equation that can be set up: \latex{\frac{4}{9}x-12=14}.
The solution:
The solution:
\latex{\frac{4}{9}x=26},
\latex{x=58.5}.
Checking: On the first day we spent \latex{\frac{1}{3}\times 58.5+9=28.5} EUR, thus \latex{ 30 } EUR remained.
On the second day we spent \latex{\frac{1}{3}\times 30+6=16} EUR out of it, thus we were left with \latex{ 14 } EUR, which we spent on the last day. These make a total of
On the second day we spent \latex{\frac{1}{3}\times 30+6=16} EUR out of it, thus we were left with \latex{ 14 } EUR, which we spent on the last day. These make a total of
\latex{ 28.5 } EUR \latex{ + } \latex{ 16 } EUR \latex{ + } \latex{ 14 } EUR = \latex{ 58.5 } EUR.
Thus we took \latex{ 58.5 } EUR with us for the trip.
Example 4
The sum of the digits of a two-digit number is \latex{ 10 }. If we swap the digits, then we get a number \latex{ 36 } greater than the original number. Give this number.
Solution
Generally a two-digit number \latex{\overline{ab}} can be written as follows: \latex{10a+b}.
Let the number of tens in the original number be \latex{ x }.
Let the number of tens in the original number be \latex{ x }.
Number of
tens
tens
The number
Original number
Number with
swapped digits
swapped digits
\latex{x}
\latex{10-x}
\latex{10x+10-x=9x+10}
\latex{10(10-x)+x=100-9x}
Number of
units
units
\latex{10-x}
\latex{x}
The equation that can be set up based on the exercise:
\latex{9x+10+36=100-9x},
\latex{18x=54},
\latex{x=3}.
According to this the original number is 37. The number with swapped digits is \latex{ 73 }, and indeed: \latex{73 - 37 = 36}.
Example 5
The ratio of two larger angles of a triangle is \latex{2 : 3}. Its smallest angle is \latex{60^{\circ}} less than the largest angle. Give the angles of the triangle.
Figure 24
\latex{\alpha}
\latex{\beta}
\latex{\gamma}
\latex{ A }
\latex{ B }
\latex{ C }
Solution
Let us denote the angles of the triangle the usual way as shown in Figure 24. Let the two larger angles be denoted by \latex{\beta} and \latex{\gamma}, and let the smallest angle be denoted by \latex{\alpha}.
According to the ratio we can set up the following equations: \latex{\beta=2x}; \latex{\gamma=3x}. Thus the smallest angle is: \latex{\alpha=\gamma-60^{\circ}=3x-60^{\circ}}.
Since the sum of the interior angles of the triangle is \latex{180^{\circ}}, therefore we can set up the following equation:
According to the ratio we can set up the following equations: \latex{\beta=2x}; \latex{\gamma=3x}. Thus the smallest angle is: \latex{\alpha=\gamma-60^{\circ}=3x-60^{\circ}}.
Since the sum of the interior angles of the triangle is \latex{180^{\circ}}, therefore we can set up the following equation:
\latex{2x+3x+3x-60^{\circ}=180^{\circ}}.
Solving this:\latex{8x-60^{\circ}=180^{\circ}},
\latex{8x=240^{\circ}},
\latex{x=30^{\circ}}.
The angles of the triangle in ascending order will be as follows: \latex{\alpha=30^{\circ}}, \latex{\beta=60^{\circ}} and
\latex{\gamma=90^{\circ}}.
\latex{\gamma=90^{\circ}}.
We can observe that during the solution it is worth adhering to the following steps:
- We look for logic relations in the text, and we select the unknown. (In most of the cases it is practical to select the unknown based on the question.)
- We set up the equation.
- We solve it.
- We check the solution against the text.
- We formulate our answer.
Exercises
{{exercise_number}}. On the first day of a three-day bicycle trip we cycled one fourth of the distance and \latex{ 6 } \latex{ km }, on the second day we covered one third of the remaining distance and \latex{ 2 } \latex{ km }, thus \latex{ 44 } \latex{ km } remained for the last day. Give the distance of the bicycle trip.
{{exercise_number}}. The sum of the ages of three brothers is \latex{ 40 } years. The middle brother is \latex{ 3 } years older than the youngest brother, but he is \latex{ 4 } years younger than the oldest. How old are they?
{{exercise_number}}. A father says the following to his \latex{ 8 }-year-old daughter: “When you will be my age, I will be \latex{ 60 } years old.” How old is the father?
{{exercise_number}}. I thought of a number. I added \latex{ 4 } to it. I multiplied the sum by \latex{ 2 }, and then I subtracted \latex{ 8 } from the result, this way I got the same number that I thought of. Give this number.
{{exercise_number}}. In a two-digit number the digit standing in the tens place is \latex{ 1 } less than three times the digit standing on the ones place. If we swap the digits, then we get a number \latex{ 27 } less. Give this number.
{{exercise_number}}. Someone says the following: \latex{ 80}% of my money produced \latex{ 5 }% interest, \latex{ 15 }% of my money produced \latex{ 3 }% interest, and \latex{ 5 }% of my money produced \latex{ 2 }% interest in the last year. The yearly interest was \latex{ 364 } EUR. How much money did he/she have in total?
Puzzle
The forest management decides to chop down pine trees from a forest. The environmental activists start a huge protest; in return the head of the management tries to calm them down with the following: “\latex{ 99 }% of the forest consists of pine trees, after the chopping \latex{ 98 }% of the forest will still be pine trees.”
What percentage of the forest do they want to chop down?
What percentage of the forest do they want to chop down?
