Linear equations and inequalities

1003031101

Level: 
C
So far Johnny has received the grades: \( 5 \), \( 3 \), \( 3 \), \( 3 \), \( 2 \), \( 2 \), \( 1 \), \( 1 \), for the math course in this school term. What another grade he needs to get so that the average of grades is lower than \( 2.5 \)? (Assumption: All grades are of the same weight in grading scale of \( 1 \), \( 2 \), \( 3 \), \( 4 \), \( 5 \), where \( 1 \) is the best result.)
at worst \( 2 \)
at worst \( 3 \)
only \( 1 \)
The arithmetic mean can’t be less than \( 2.5 \) in any case.

1003029706

Level: 
C
In a river, water flows at speed of \( 1\,\mathrm{mps} \). A boat, which moves at speed of \( 4\,\mathrm{mps} \) in a calm water, carries mail to a little town distant \( 6\,\mathrm{km} \) downstream. How long will it take till the boat returns? (We disregard the time necessary for mail-handover.) Note: Abbreviation "mps" stands for meter per second.
\( 53\,\mathrm{min}\ 20\,\mathrm{s} \)
\( 50\,\mathrm{min} \)
\( 3\,\mathrm{min}\ 12\,\mathrm{s} \)
\( 1\,\mathrm{min}\ 20\,\mathrm{s} \)

1003029704

Level: 
C
Tom is \( 15 \) years old and his grandpa is \( 67 \). How many years will it take for the grandpa to be three times as old as Tom? Choose the equation which represents the algebraic solution to the given word problem.
\( 67+x=3\cdot(15+x) \)
\( 67=3\cdot x \)
\( 67=3\cdot(15+x) \)
\( 3\cdot(67+x)=3\cdot(15+x) \)

1003029703

Level: 
C
A sample of \( 168 \) respondents of a sociological research splits into three categories according to nationality: Czech, Slovak and other. There is four times less Slovak respondents than Czech in the sample. The other nationality reported \( 85\% \) less respondents than Czech nationality. Choose the equation that represents the algebraic relation between the nationalities of the sample and leads to evaluation of the number of people of the given nationality.
\( x+4x+\frac{4x\cdot15}{100}=168 \), where \( x \) is the number of people of the Slovak nationality
\( x+\frac x4+\frac{x\cdot15}{4\cdot100}=168 \), where \( x \) is the number of people of the Slovak nationality
\( x+4x+\frac{4x\cdot85}{100}=168 \), where \( x \) is the number of people of the Slovak nationality
\( x+4x+\frac{4x\cdot15}{100}=168 \), where \( x \) is the number of people of the Czech nationality

1003029702

Level: 
C
The comparative math test consists of the total of three problems. From students taking the test \( 16\% \) solved correctly all three problems, \( 3/5 \) of the students made mistake in exactly one of the problems and \( 2/15 \) of the students made mistakes in exactly two of the problems. Eight students gave neither one of the problems. How many students wrote the test without a mistake?
\( 12 \)
\( 75 \)
\( 45 \)
\( 10 \)

1003029701

Level: 
C
One and a half of a loaf of bread weighs as same as a quarter of a loaf of bread together with a kilo weight. If it is possible, find how much one loaf weighs. (Assume that all the loaves weigh the same.)
\( 0.8\,\mathrm{kg} \)
\( 1.25\,\mathrm{kg} \)
\( 0.2\,\mathrm{kg} \)
It is not possible to determine the exact weight of the loaf from the given information.

1003037502

Level: 
B
Five more than one-third of the unknown number is smaller than three times the difference of this number and five. Which numbers do satisfy the given condition?
greater than \( \frac{15}2 \)
greater than \( \frac{15}4 \)
greater than \( \frac{25}4 \)
greater than \( \frac52 \)

1003037501

Level: 
B
The difference of one-fifth of the unknown number and one-half of the number is greater than the difference of the number and thirteen. How many natural numbers do satisfy the given condition? (Note: Natural numbers are corresponding to positive integers.)
\( 9 \)
\( 10 \)
infinity
\( 18 \)