Sommaire de cette fiche
 1 Excerpts from the patent on numerical calculation
 2 Systems of two equations with two unknowns
 3 Arithmetic
 4 Literal calculation and product equation
 5 Problem on literal calculation
 6 Factoring and product equation
 7 Develop and literal calculation
 8 Book cover
 9 Solving equations
 10 Filling rate of a box and geometry in space
 11 Square roots
 12 Exercise on the gearshift knob
 13 Frequency and statistics
 14 Percentages and graphical representation
 15 Histogram and frequency
 16 Calculation of mean and median
 17 Statistics and cinema
 18 Mean, median and range
 19 Calculation of average
 20 Park of the cité des sciences with the geode
 21 Exercise on a well
 22 Pythagorean reciprocal and area
 23 Theorem of Thales and Pythagoras
 24 Thales and Pythagoras on a rectangular field
 25 Trigonometry and circumscribed circle
 26 Theorem of Thales
 27 Trigonometry and geometric figure
 28 Arithmetic and rectangular field
 29 Dimensions of a case
 30 Cans
 31 Coordinates of a point
 32 Square roots and integer
 33 Simplification of square roots
 34 Calculation of percentages
 35 Box of chocolates and geometry in space
 36 Wooden ball and geometry in space
 37 Aquarium
 38 Systems of two equations with two unknowns
 39 Financing a trip
 40 Chickens and ducks
 41 Book cover
 42 A factory tests light bulbs
 43 Number of vehicles sold
 44 Literal calculation and right triangle
Excerpts from the 2023 math patent topics organized by chapter.
These extracts allow you to revise the brevet des collèges in order to prepare yourself in the best conditions.
In addition to all the subjects of the brevet de maths of the previous sessions, Mathovore provides you with extracts of subjects which target each chapter of the program of third (3ème).
Excerpts from the patent on numerical calculation
Exercise 1:
Calculate and give the result as an irreducible fraction:
.
Exercise 2:
Calculate and give the result in scientific notation:
Exercise 3:
1. We give:
.
Calculate A and give the result as a fraction.
2. We give .
Write B in the form where b is an integer .
3. We give
Calculate C and give the scientific form of the result.
Exercise 4:
Calculate and put into the simplest form possible :
.
.
.
.
Systems of two equations with two unknowns
Consider the following system of equations of the first degree: :
.
1. Is the couple (x=2;y=0.5) a solution of this system ?
2. Solve this system of equations.
3. At the bakery, Anatole buys 2 croissants and 3 pains au chocolat : he pays 5,50 €.
Beatrice buys 3 croissants and 1 pain au chocolat and pays 4,05 € .
What is the price of a croissant? What is the price of a pain au chocolat?
Arithmetic
1. Are the numbers 682 and 352 prime between them ? Justify.
2. Calculate the greatest common divisor (GCD) of 682 and 352.
3. Make the fraction irreducible by clearly indicating the method used.
Literal calculation and product equation
We give: .
1. Show, by detailing the calculations, that D can be written: D = (2x – 3)(7x + 1)
2. Solve the equation: (2x – 3)(7x + 1) = 0.
Literal calculation and factoring
1. Let.
a. Which remarkable identity allows you to factor D?
b. Factorize D.
2. Let
a. Develop E.
b. Factorize E.
c. Solve the equation: 6x(3x + 1) = 0.
Problem on literal calculation
1. Eric says to Zoe, “Pick a number x; add 1 to the triple of x; then calculate the square of the resulting number and subtract the number 4 from it.”
What result will Zoe find if she chooses: x = 5?
2. Eric offers Zoe four expressions, one of which corresponds to the calculation he had her do.
These four expressions are:
Which expression should Zoë choose?
3. a. Factor: .
b. Solve: (3x – 1)(3x + 3) = 0.
c. Zoe plays again; she chooses a negative number and finds zero. What number did she choose? Then check Zoe’s calculation.
Factoring and product equation
A = (2x – 3)(2x + 3) – (3x + 1)(2x – 3)
1. Expand then reduce A.
2. Factorize A.
3. Solve the equation: (2x – 3)(x + 2) = 0
Develop and literal calculation
We give
1. Develop and reduce
2. Show that
3. Find the values of x for which F = 125.
Book cover
On the cover of a geometry book are drawn figures; these are triangles or rectangles that have no common vertex.
1. How many vertices would there be if there were 4 triangles and 6 rectangles, or 10 figures in all?
2. In fact, 18 figures are drawn and we can count 65 vertices in all. How many triangles and rectangles are there on this book cover?
Solving equations
Solve each of the two equations:
3(5 + 3x) – (x – 3) = 0
3(5 + 3x)(x – 3) = 0
Filling rate of a box and geometry in space
In a cubic box whose edge measures 7 cm, we place a ball of 7 cm of diameter (see the diagram).
The volume of the ball is a certain percentage of the volume of the box. This percentage is called “box fill rate”.
Calculate the filling rate of the box.
Round this percentage to the nearest whole number.
Exercise 1:
We consider the expression :
.
1. Expand and reduce the expression E .
2. Factorize E .
3. Calculate the value of E for x = – 2 .
Solve the equation (3x+2)(5x3)=0 .
Are the solutions of this equation decimal numbers?
Exercise 2:
1. Calculate A and B, giving the result as irreducible fractions.
.
2. Consider the expression :
a. Expand and reduce C .
b. Factorize the expression C .
c. solve the equation: (2x5)(2x)=0 .
Exercise 3:
1.a. Expand and reduce the expression: D = (2x+5)(3x1) .
b. Expand and reduce the expression: E=(x1)²+x²+(x+1)² .
Application: determine three consecutive positive integers, (x1), x and (x+1) whose sum of squares is 4 802 .
2.a. Factor the expression: F=(x+3)²(2x+1)(x+3) .
b. Factor the expression: G=4x²100 .
Application: determine a positive number whose square of the double is equal to 100 .
Exercise 4:
1. Factorize :
a. 912x+4x² .
b. (32x)²4 .
2. Deduce a factorization of : E = (912x+4x²)4 .
Exercise 5:
We pose E=(4x3)²+6x(4x)(x²+9).
a. Show that E is equal to the square of 3x .
b. Find the values of x for which E=144 .
c. Calculate the value of E for
Square roots
Exercise 1:
We pose .
Write E in the form .
( a and b being relative numbers) .
Exercise 2:
Calculate D and E and give the results in the form where a and b are integers with b as small as possible.
Exercise 3:
We give:
Write A in the form and B in the form where a and b are two relative integers .
Exercise 4: (Foreign centers)
We pose:
1. Calculate a², b² and a²+b² .
2. Show that a²+b² is an integer.
3. If a and b are the lengths of the sides of the right angle in a triangle, what is the length of the hypotenuse?
Exercise on the gearshift knob
Figure 1 shows the gearshift knob of an automobile.
It has the shape of a halfball on top of a cone whose end has been cut off as shown in figure 2 .
We call the cone whose base is the circle of radius [AH] and the cone whose base is the circle of radius [EK].
These two circles are located in parallel planes.
– Reminder of the formulas :
Volume of a cone:
Volume of a ball :
We pose : Sk = 4 cm ; SH=10 cm ; AH = 2 cm .
1. Given the triangle SAH, calculate the tangent of the angle .
Deduce an approximate value, to the nearest degree, of the angle .
2. By placing in the rightangled triangle ESK and using the tangent of the angle , show that: EK= 0.8 cm .
3.a. Calculate the volumes and of the cones and .
Approximate values will be given for the two requested volume calculations to the nearest .
b. Calculate the volume of the halfball; give an approximate value to the nearest .
c. Deduce from the previous results an approximate value of the volume of the pommel .
Frequency and statistics
Here is a table giving the population of French Polynesia by age group in 1996.
1. Complete the table below.
Frequencies will be expressed as percentages, rounded to the tenth.
2. Calculate the number of people who are under 40 years of age.
3. Calculate the number of people aged 40 years or older.
Percentages and graphical representation
On the occasion of the final of the women’s world handball championships, the regional daily newspaper “Le télégramme” titled on 16/12/1999: “the Breton hand more feminine than the French hand”.
The data are as follows:
Age 
[0 ; 20[ 
[20 ; 40[ 
[40 ; 60[ 
60 and over 
Total 
Workforce 
94 651 
75 537 
37 940 
13 193 

Frequency 
Brittany 
France 

Licensees 
15 350 
230 000 
(Of which) women 
6 600 
87 000 
a. Calculate the percentage of women among the licensees in Brittany, then the percentage of women among the licensees in France. (Rounding to the nearest whole number will be given).
b. Make a graphical representation that will highlight the phenomenon highlighted in the title.
Histogram and frequency
At the exit of a city, the distribution of the 6400 vehicles leaving the city between 4 p.m. and 10 p.m. was recorded on a certain day. The results are given in the table below:
Time slot 
16h 17h 
17h 18h 
18h 19h 
19h 20h 
20h 21h 
21h 22h 
Number of vehicles 
1 100 
2 000 
1 600 
900 
450 
350 
1. Show the histogram of the numbers of this statistical series.
2. Calculate the frequency of the time slot 19h20h (we will give the result rounded to the nearest 0.01, then the corresponding percentage).
3. Calculate the percentage of vehicles leaving the city between 4:00 and 8:00 p.m.
Calculation of mean and median
In meteorology, the number of hours of sunlight is called insolation.
Here are the weather reports for Voglans in Savoie giving information about the sunshine in July of the last years.
Years 
1990 
1991 
1992 
1993 
1994 
1995 
1996 
1997 
1998 
1999 
2000 
Insolation (h) 
324 
325 
257 
234 
285 
261 
213 
226 
308 
259 
206 
1. Calculate the average insolation over this period (the result will be rounded to the nearest hour)
2. Can we say that the value 259 is the median of this series ? Justify.
Statistics and cinema
Students compared prices at 5 different movie theaters. They each took a few friends to a movie theater and recorded their spending in the following table:
Cinema A 
Cinema B 
Cinema C 
Cinema D 
Cinema E 

Number of seats purchased 
3 
5 
7 
4 
6 
Amounts spent (in €) 
16,02 
25 
42,70 
24,80 
1. The student who went to cinema E lost the ticket, but he knows that the price was the same as in cinema D. Calculate the price paid by this student for the 6 seats purchased.
2. a. Determine which theater has the cheapest fare.
b. Calculate, in euros, the average of the prices charged.
Mean, median and range
Here is the series, ordered in ascending order, of the 15 grades obtained in mathematics by a student during the first semester:
4 – 6 – 6 – 9 – 11 – 11 – 12 – 13
13 – 13 – 14 – 15 – 17 – 18 – 18
1. What is the frequency of the note 13 ?
2. What is the average score?
3. What is the median score?
4. What is the range of this series of notes?
Calculation of average
1. The marks obtained by the students of a^{3rd} grade class on a mathematics assignment are given in the table below:
Notes 
7 
8 
8,5 
9 
10 
11 
13 
15,5 
18 
Workforce 
1 
2 
2 
4 
4 
6 
3 
2 
1 
Calculate the class average by detailing the calculations on the copy.
2. Students in another class have a transcript that matches the table below:
Notes 
7,5 
8,5 
9 
10 
10,5 
11 
13 
14 
16 
17 
Workforce 
2 
3 
4 
4 
1 
5 
3 
3 
1 
1 
Determine a median value of this series of scores, justify.
Percentage and average age
The table below shows the age distribution of students in the middle school dugout club.
Age of students 
11 
12 
13 
14 
Number of students 
4 
7 
10 
3 
1. Calculate the club’s total membership.
2. Calculate the average age of students in the club.
3. Calculate the percentage of students under the age of 14 in this club.
Park of the cité des sciences with the geode
In the park of the Cité des Sciences is the Géode, a movie theater which has, externally, the shape of a spherical cap placed on the ground, with a radius of 18 m.
1. Calculate OH .
2. Calculate HM (give the result rounded to the nearest 1 m).
3. calculate the total height of the geode .
4. a. What is the shape of the ground surface occupied by the geode?
b. Calculate the area of this surface (round the result to the nearest 1 m²).
5. We want to represent the triangle OMH to scale .
a. What is the length Om on this representation?
Construct the triangle OMH to the scale .
Exercise on a well
[AD] is a diameter of a cylindrical well.
Point C is at the vertical of D, at the bottom of the well.
A person places himself at a point E on the halfline [DA) so that his eyes are aligned with points A and C.
We note Y the point corresponding to the eyes of this person.
We know that:
AD = 1.5 m; EY=1.7 m; EA=0.6 m .
Prove that the lines (DC) and (EY) are parallel.
2. Calculate DC, the depth of the well.
Right triangle, circumscribed circle and Pythagoras
Let [IJ] be a segment of length 8 cm.
On the circle (C) of diameter [IJ], consider a point K such that IK = 3.5 cm.
1. Make the figure.
2. Prove that the triangle IJK is rightangled.
3. Calculate JK (the result should be rounded to the nearest mm).
Pythagorean reciprocal and area
The figure below is not full size.
We give the following lengths in cm: BH= 5.8 cm; HC = 4.5 cm; AC = 7.5 cm; AH = 6 cm.
1. Using only a ruler and a compass, construct this figure in true size (leave the construction lines visible).
2. Show that the triangle ACH is rightangled at H.
3. Calculate the area of triangle ABC.
4. Let M be the midpoint of [AC] and D the symmetric of H with respect to M.
Place M and D on the figure made in question 1.
Show that the quadrilateral ADCH is a rectangle.
Theorem of Thales and Pythagoras
Consider the figure below:
We give MN = 8 cm; ML = 4.8 cm; LN = 6.4 cm.
We do not ask you to redo the figure on the copy.
1. Prove that the triangle LMN is rightangled.
2. Let S be the point of [MN] such that NS = 2 cm.
The perpendicular to (LM) passing through S intersects [LN] at R.
Calculate RS.
Thales and Pythagoras on a rectangular field
The figure below represents a rectangular field ABCD crossed by a road of uniform width (gray part).
We give:
– AB = 100 m BC = 40 m AM = 24 m
– The lines (AC) and (MN) are parallel.
Calculate:
1. The value of the length AC, rounded to the nearest decimeter.
2. MB length.
3. The length BN.
Trigonometry and circumscribed circle
We call (C) the circle of center O and diameter [AB] such that: AB = 8cm.
M is a point of the circle such that: .
1. Make the figure in full size.
2. What is the nature of the BAM triangle? Justify.
3. Calculate the length BM rounded to the nearest 0.1 cm.
Theorem of Thales
In the figure below, drawn freehand:
IR = 8 cm RP = 10 cm IP = 4 cm
IM = 4 cm IS = 10 cm IN = 6 cm IT = 5 cm
We don’t ask you to do it again.
1. Show that the lines (ST) and (RP) are parallel.
2. Deduce ST.
3. Are the lines (MN) and (ST) parallel ? Justify.
Trigonometry and geometric figure
The figure is not to scale.
Consider the circle (C) with center O, point of the halfline [Ay). The halfline [Ax) is tangent to (C) at T. We give AT = 9 cm.
1. Calculate an approximate value, to the nearest millimeter, of the radius of circle (C).
2. At what distance from A must a point B be placed on [AT] so that the angle measures ?
(Give an approximate value rounded to the nearest millimeter).
Arithmetic and rectangular field
1. Compute PGCD(39; 135).
2. Christopher has a rectangular field that he wants to fence off. The dimensions of the field are, in meters, 39 by 135. He wants to plant poles at a regular distance greater than 2 m and measured by a whole number in meters. In addition, he places a post at each corner.
a. What is the distance between two posts?
b. How many poles should he plant?
Dimensions of a case
The dimensions of a box are 105 cm, 165 cm and 105 cm. We want to make cubic boxes, as big as possible, which allow to fill the box completely.
What should be the edge of these boxes and how many such boxes can be placed in the box?
Covered floor
A rectangular room measures 4.2 m by 8.7 m. Its floor is covered with whole and square tiles.
1. What is the largest possible size for each of these slabs?
2. How many of these tiles are needed to cover the floor of the room?
Cans
We have two cans of 18 liters and 15 liters respectively. By pouring an integer number of times the contents of a container into each of them, they can be filled exactly.
What is the largest possible capacity of this container?
Soccer clubs
With little money, two soccer clubs decided to merge. The former has 120 members and the latter 144.
To define the terms of the merger, a committee is formed. The number of representatives from each club shall be in proportion to the number of members. We would like the committee to be as small as possible.
How many representatives should each club designate?
Coordinates of a point
We join the origin of the reference frame O to the point A of coordinates (72; 48).
1. Through how many points whose two coordinates are integers does the segment pass?
2. Give the coordinates of these points.
Square roots and integer
We consider the number :
Write B as an integer.
Simplification of square roots
Calculate:
We will give the result in the form where a is an integer.
Calculation of percentages
The General Council of a department has 60 elected members. Each of them represents one of the three parties A, B and C.
– Party A has 15 elected members;
– 45% of the elected officials belong to party B ;
– the rest of the elected representatives are from party C.
1. Calculate the percentage of elected officials who belong to party A.
2. Calculate the number of elected officials of party B.
3. Show the distribution of the General Council between parties A, B and C in a pie chart of radius 4 cm.
Box of chocolates and geometry in space
A box of chocolates has the shape of a regular pyramid with a square base, cut by a plane parallel to the base.
The upper part is the lid and the lower part contains the chocolates.
We give: AB = 30 cm SO = 18 cm SO’ = 6 cm
1. Calculate the volume of the pyramid SABCD.
2. Deduce that of the SEFGH pyramid.
3. Calculate the volume of the container ABCDEFGH that contains the chocolates.
Wooden ball and geometry in space
A carpenter has to cut wooden balls of 10 cm in diameter to put them on a staircase banister.
He first makes 10 cm cubes from which he cuts each ball.
1. Using only the data in the statement, draw a fullsize triangle OHA, rightangled at H.
The construction lines will be left visible.
2. Calculate the radius of the circle (C).
Aquarium
An aquarium has the shape of a spherical cap of center O (see diagram below), which has for radius R = 12 and for author h = 19,2 (in centimeters).
1. Calculate the length OI and then the length IA.
2. The volume of a spherical cap is given by the formula :
where R is the radius of the sphere and h is the height of the spherical cap.
Calculate the approximate value of the volume of this aquarium to the nearest ^{cm3}.
3. We pour 6 liters of water into this aquarium.
When changing the water in the aquarium, the contents are transferred into a parallelepipedic container 26 cm long and 24 cm wide.
Determine the height x of the water in the container. Round the result to the nearest mm.
Systems of two equations with two unknowns
1. Solve the following system:
2. The difference of two numbers is 24.
What are these two numbers knowing that if we increase both of them by 8 ,
we obtain two new numbers whose largest is the triple of the smallest?
Financing a trip
1. Solve the following system of two equations with two unknowns:
2. To finance part of their endofyear trip,
third graders sell cakes they have made themselves.
On one day, they sold 15 pies, some blueberry and some apple.
A blueberry pie is sold for 4 euros and an apple pie for 2 euros.
The amount collected that day is 42 euros.
After putting the problem into an equation, determine how many of each kind of pie they sold.
Chickens and ducks
A farmer sells 3 ducks and 4 chickens for 70,30 €.
A duck and a chicken are worth 20,70 € together.
Determine the price of a chicken and a duck.
Book cover
On the cover of a geometry book are drawn figures ;
these are triangles or rectangles that have no common vertex.
1. How many vertices would there be if there were 4 triangles and 6 rectangles, or 10 figures in all?
2. In fact, 18 figures are drawn and we can count 65 vertices in all.
How many triangles and rectangles are there on this book cover?
A factory tests light bulbs
A factory tests light bulbs, on a sample, by studying their life span in hours.
Here are the results:
d : lifetime in hours 
Number of bulbs 
1,000 < d < 1,200 
550 
1,200 < d < 1,400 
1 460 
1,400 < d < 1,600 
1 920 
1,600 < d < 1,800 
1 640 
1,800 < d < 2,000 
430 
1. What percentage of bulbs have a life of less than 1400 h?
2. Calculate the average life of a bulb.
Number of vehicles sold
In the year 2000, the number of new cars sold in France was 2,134 thousand, distributed as follows:
– 602 thousand Renault ;
– 262 thousand Citroën ;
– 398 thousand Peugeot ;
– cars of foreign brands.
1. What is the sales frequency, expressed as a percentage and rounded to 1%, for foreignbranded cars?
2. In the total sales of French cars, what percentage do Renault cars represent?
Literal calculation and right triangle
1. Expand and reduce the expression :
2.factor the expression :
3. ABC is a rightangled triangle at A; x is a positive number; BC = x + 7; AB = 5.
Make a diagram and show that :
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