Systems of linear equations and inequalities

1103034507

Level: 
B
Consider a balance scales comprising of beam with unequal length of arms where the fulcrum is very close to one end of the beam. (Such scales are called steelyard. For instance, it is often used to weigh a catch in fisheries.) The load is hanged on the shorter arm, while the balance about the fulcrum is obtained by sliding the counterweight along the longer arm. (See the picture.) Suppose the distance of the load hanging point from the fulcrum is fixed at \( 5\,\mathrm{cm} \). If the weight of the load is \( 80\,\mathrm{N} \), the balance is achieved as the counterweight is moved to the very end of the longer arm. If the weight of the load is \( 60\,\mathrm{N} \), the balance is achieved when the counterweight is moved to the distance of \( 30\,\mathrm{cm} \) from the fulcrum. What is the length of the beam? \[ \] Hint: The steelyard is based on the law of the lever. For balanced lever is: \( F_1\cdot a=F_2\cdot b \), where \( F_1 \) is the weight of the load in the distance \( a \) from the fulcrum and \( F_2 \) is the weight of the counterweight in the distance \( b \) from the fulcrum.
\( 45\,\mathrm{cm} \)
\( 54\,\mathrm{cm} \)
\( 40\,\mathrm{cm} \)
\( 35\,\mathrm{cm} \)

1003034506

Level: 
B
Kamil is able to mow a meadow in \( 12 \) hours. Zdeněk has a better lawn mower and he is able to mow the same meadow in \( 8 \) hours. They have agreed that Kamil starts to mow alone sooner and Zdeněk will join him later so that the total time of mowing is \( 9 \) hours. How long will they mow together?
\( 2 \) hours
\( 7 \) hours
\( 6 \) hours
\( 3 \) hours

1003034505

Level: 
B
The March price of a T-shirt and shorts was \( 600\,\mathrm{CZK} \) together. In April there was on store price adjustment. The price of the shorts decreased by \( 10\% \) and the price of the T-shirt increased by \( 10\% \). So the April price of both together the shorts and the T-shirt was by \( 20\,\mathrm{CZK} \) lower. What was the April price of the T-shirt?
\( 220\,\mathrm{CZK} \)
\( 200\,\mathrm{CZK} \)
\( 180\,\mathrm{CZK} \)
\( 400\,\mathrm{CZK} \)

1003034503

Level: 
B
Students registered for sports camps. For the biking camp registered by \( 18 \) students more than for the boating camp. After some time one of the students switched his registration from the boating camp to the biking camp. Now, there is two times more bikers than boaters. How many students registered originally for the boating camp?
\( 21 \)
\( 39 \)
\( 20 \)
\( 15 \)

1003034502

Level: 
C
Petr would like to buy a new smartphone. If he starts a temporary job at the electro shop he gets payed \( 120\,\mathrm{CZK} \) per hour plus he gets \( 20\% \) discount on the smartphone bough in the shop. He calculated that for \( 24 \) hours of work he would not earn even half of the phone price. Another employer pays \( 150\,\mathrm{CZK} \) per hour. If Petr gets a temporary job with the other employer he is not eligible for the discount in the electro shop anymore, however he can buy the smartphone from e-shop for the price by \( 600\,\mathrm{CZK} \) lower than in the electro shop and for \( 20 \) hours of work he earns more than one third of the electro shop smartphone price. Determine as closely as possible the smartphone price in the electronics shop.
greater than \( 7\,200\,\mathrm{CZK} \) and less than \( 9\,600\,\mathrm{CZK} \)
greater than \( 7\,200\,\mathrm{CZK} \) and less than \( 10\,800\,\mathrm{CZK} \)
greater than \( 4\,800\,\mathrm{CZK} \) and less than \( 9\,600\,\mathrm{CZK} \)
greater than \( 4\,800\,\mathrm{CZK} \) and less than \( 10\,800\,\mathrm{CZK} \)

1003034501

Level: 
C
Two different aquarium fish shops have special price on Congo Tetra. The price is \( 42\,\mathrm{CZK} \) for one fish. Further, the discount of \( 50\,\mathrm{CZK} \) from a purchase over \( 300\,\mathrm{CZK} \) is offered in shop A. In shop B a customer is given \( 5\% \) discount of the price of any purchase. How many of Congo Tetra must one buy to make the total price in shop A lower than the total price in shop B?
greater than \( 7 \) and less than \( 24 \)
less than \( 24 \)
greater than \( 23 \)
less than \( 7 \)

1003083003

Level: 
A
Find the solution set of the following system of equations. \[ \begin{aligned}\frac23 x-\frac12y&=1 \\ -2x+\frac32y&=-3 \end{aligned} \]
\( \left\{\left[x; \frac{4x-6}3\right]\colon x\in\mathbb{R}\right\} \)
\( \left\{\left[x; y\right]\colon x\in\mathbb{R}\text{, } y\in\mathbb{R}\right\} \)
\( \emptyset \)
\( \left\{[0; -2]\right\} \)

1003083002

Level: 
A
Identify which of the sets is not the solution set of the following system of equations. \[ \begin{aligned} \frac12 x-y&=3 \\ \frac x3 - \frac23 y &=2 \end{aligned} \]
\( \left\{\left[6+2y;\frac{x-6}2\right]\colon x\in\mathbb{R}\text{, }y\in\mathbb{R}\right\} \)
\( \left\{\left[x; \frac{x-6}2\right]\colon x\in\mathbb{R}\right\} \)
\( \left\{\left[6+2y;y\right]\colon y\in\mathbb{R}\right\} \)
\( \left\{\left[2t;t-3\right]\colon t\in\mathbb{R}\right\} \)