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Numbers and operations
Example 1
By keeping the order of the numbers \latex{ 1;\, 2;\, 3;\, 4;\, 5; } place the symbols of the four basic operations, and, if necessary, parentheses between them such that the resulting natural number is
- \latex{0;}
- \latex{1.}
Solution (a)
A couple of possible solutions:
\latex{(1+2)\times 3-4-5=0;}
\latex{(1+2-3)\times 4\times 5=0;}
\latex{(1+2-3)\times(4+5)=0.}
\latex{(1+2-3)\times 4\times 5=0;}
\latex{(1+2-3)\times(4+5)=0.}
Solution (b)
A couple of possible solutions:
\latex{1-2+3+4-5=1;}
\latex{1+2-3-4+5=1;}
\latex{1\times (2-3)\times 4+5=1.}
\latex{1+2-3-4+5=1;}
\latex{1\times (2-3)\times 4+5=1.}
Try to find other solutions!
\latex{\N}
\latex{\Z}
\latex{\R}
\latex{\Q}
Figure 1
Example 2
Find positive integers \latex{ a } and \latex{ b } such that \latex{a+b, a\times b, \frac{a}{b}} and the geometric mean of \latex{ a } and \latex{ b } are all integers.
Solution
Since the sum and the product of any two integers are integers as well, the first two criteria are satisfied for any positive integers \latex{ a } and \latex{ b }.
\latex{\frac{a}{b}} is integer if \latex{ a } is a multiple of \latex{ b }. Let \latex{a=k\times b} where \latex{ k } is also a positive integer. Their geometric mean,
\latex{\frac{a}{b}} is integer if \latex{ a } is a multiple of \latex{ b }. Let \latex{a=k\times b} where \latex{ k } is also a positive integer. Their geometric mean,
\latex{\sqrt{a\times b}=\sqrt{k\times b^{2} }}
exists since \latex{ k } is positive, and
\latex{\sqrt{k\times b^{2} }=|b|\times \sqrt{k}=b\times \sqrt{k},}
because \latex{ b } is positive. The last expression is integer if \latex{ k } is a square number.
Therefore every pair of positive integers \latex{ a }, \latex{ b } is a solution for which the quotient of \latex{ a } and \latex{ b } is a square number.
Therefore every pair of positive integers \latex{ a }, \latex{ b } is a solution for which the quotient of \latex{ a } and \latex{ b } is a square number.
Geometric mean:
\latex{\sqrt{a\times b}; a,b\geq 0}
\latex{\sqrt{a\times b}; a,b\geq 0}
\latex{ a }
\latex{ b }
\latex{ 8 }
\latex{ 12 }
\latex{ 27 }
\latex{ 48 }
\latex{ 2 }
\latex{ 3 }
\latex{ 3 }
\latex{ 3 }
Example 3
Which digit is the \latex{ 2,004 }th after the decimal point in the decimal form of \latex{\frac{3}{7}} ?
Solution
Perform the Euclidean division:
\latex{3\div 7=04285714...}
\latex{30}
\latex{30}
\latex{20}
\latex{60}
\latex{40}
\latex{50}
\latex{10}
\latex{30}
...
\latex{30}
...
\latex{\frac{3}{7}= 0.\dot{4}2857\dot{1},} which means that there are six digits repeating in the infinite decimal form of the fraction.
Since \latex{ 6 } is a divisor of \latex{ 2,004 }, the digit in question will be the last digit of the repeating sequence: \latex{ 1 }.
Since \latex{ 6 } is a divisor of \latex{ 2,004 }, the digit in question will be the last digit of the repeating sequence: \latex{ 1 }.
Example 4
- Is it possible for two rational numbers to have an irrational arithmetic mean?
- Is it possible for two irrational numbers to have a rational arithmetic mean?
- Is it possible for two rational numbers to have an irrational geometric mean?
- Is it possible for two irrational numbers to have a rational geometric mean?
Solution (a)
All rational numbers can be written as a quotient of two integers. The sum of two fractions is a fraction itself, and by dividing with \latex{ 2 } we receive a fraction once again.
Therefore a quotient of two rational numbers cannot be irrational.
Therefore a quotient of two rational numbers cannot be irrational.
Solution (b)
It is possible. For example, \latex{\sqrt{2}} and \latex{1-\sqrt{2}} are irrational numbers, and their arithmetic mean is \latex{\frac{1}{2}.}
Solution (c)
It is possible. For example, the geometric mean of \latex{ 1 } and \latex{ 2 } is \latex{\sqrt{2}.}
Solution (d)
As one solution, take irrational numbers which are reciprocals of each other. For example, \latex{2-\sqrt{3}} and \latex{2+\sqrt{3}} will do as their product is:
\latex{(2-\sqrt{3} )\times (2+\sqrt{3} )=4-(\sqrt{3} )^{2}=4-3=1.}
Therefore the geometric mean of two irrational numbers can be rational.
Example 5
A company operates two plants. Each day the first one produces \latex{ 2,400 } and the second one produces \latex{ 1,500 } units of a product. How many percents will the company’s production increase if the first plant increases its daily output by \latex{ 15 }% and the second one by \latex{ 10 }%?
Solution
Compute how much the daily production of each plant will increase. The first plant will produce \latex{2,400\times \frac{15}{100}=360} units more, while the second plant will manufacture \latex{1500\times \frac{10}{100}=150} units more every day.
The company's output increases by \latex{360+150=510} units per day.
This is \latex{\frac{510}{3,900}=0.1308} part of the original output. Therefore, production volume of the company will be increased by approximately \latex{13\%.}
This is \latex{\frac{510}{3,900}=0.1308} part of the original output. Therefore, production volume of the company will be increased by approximately \latex{13\%.}
Exercises
{{exercise_number}}. Which digit is the \latex{ 1,960 }th after the decimal point in the decimal form of \latex{\frac{8}{17}?}
{{exercise_number}}. Decide whether it is true that \latex{\sqrt{2}+\sqrt{3}} is an irrational number or not. Verify your answer.
{{exercise_number}}. Find irrational numbers all of whose digits are either \latex{ 2 } or \latex{ 3 }.
{{exercise_number}}. A tourist map is scaled \latex{ 1:40,000 }. How much do we have to walk if our destination is \latex{ 5\, cm } away from our position on the map?
{{exercise_number}}. Price of a coat was increased by \latex{ 20 }% at the beginning of the year, and then decreased by \latex{ 20 }% during the end of season sale. What percentage of the original price does it cost now?
{{exercise_number}}. A merchant would earn a profit of \latex{ 30 }% if they sell a product for \latex{ 10,920 } Euros. Since the product was not selling at all, its price was decreased by \latex{ 10 }%. What is the profit afterwards?
{{exercise_number}}. Packaging of a food product reads as follows: gross weight is \latex{ 4,850\, g }, net weight is \latex{ 3,750\, g }. How many percent is the net weight of the gross? How many percent is the packaging of the net weight?
{{exercise_number}}. Three trips were organized for a class during the school year. The first, the second and the third one was attended by \latex{ 70 }%, \latex{ 80 }% and \latex{ 90 }% of the students, respectively. \latex{ 12 } students attended all three trips, while the others had attended two. How many students are there in the class?