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Solving Exercises FINAL Chapter 6 Math 7 Connect
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Solve lesson 6.33 on page 21 Math 7 textbook Connecting knowledge volume 2
Make all possible proportions from the following four numbers: 0.2; 0.3; 0.8; 1,2.
Solution method
Step 1: Find the equality obtained from the above 4 numbers.
Step 2: With ad= bc (a,b,c,d \( \ne \) 0), we have the following proportions:
\(\dfrac{a}{b} = \dfrac{c}{d};\dfrac{a}{c} = \dfrac{b}{d};\dfrac{d}{b} = \dfrac{ c}{a};\dfrac{d}{c} = \dfrac{b}{a}\)
Detailed explanation
We have: 0.2 . 1.2 = 0.3 . 0.8
The possible ratios are:
\(\dfrac{{0,2}}{{0,3}} = \dfrac{{0,8}}{{1,2}};\dfrac{{0,2}}{{0.8) }} = \dfrac{{0,3}}{{1,2}};\dfrac{{1,2}}{{0,3}} = \dfrac{{0,8}}{{0, 2}};\dfrac{{1,2}}{{0,8}} = \dfrac{{0,3}}{{0,2}}\)
Solve lesson 6.34 page 21 Math 7 Textbook Connecting knowledge volume 2
Find the unknown element x in the aspect ratio: \(\dfrac{x}{{2,5}} = \dfrac{{10}}{{15}}\)
Solution method
Apply the property of proportions: If \(\dfrac{a}{b} = \dfrac{c}{d}\) then ad = bc
Detailed explanation
Since \(\dfrac{x}{{2,5}} = \dfrac{{10}}{{15}}\) then x. 15 = 2.5 . 10 \( \Rightarrow 15.x = 25 \Rightarrow x = \dfrac{{25}}{{15}} = \dfrac{5}{3}\)
So \(x = \dfrac{5}{3}\)
Solve lesson 6.35 on page 21 Math 7 textbook Connecting knowledge volume 2
From the proportions \(\dfrac{a}{b} = \dfrac{c}{d}\) (with a,b,c,d other than 0) what proportions can be inferred?
Solution method
Apply the property of proportions: If \(\dfrac{a}{b} = \dfrac{c}{d}\) then ad = bc
With ad= bc (a,b,c,d \( \ne \) 0), we have the following proportions:
\(\dfrac{a}{b} = \dfrac{c}{d};\dfrac{a}{c} = \dfrac{b}{d};\dfrac{d}{b} = \dfrac{ c}{a};\dfrac{d}{c} = \dfrac{b}{a}\)
Detailed explanation
We have: \(\dfrac{a}{b} = \dfrac{c}{d}\) so ad = bc
We get the proportions: \(\dfrac{a}{c} = \dfrac{b}{d};\dfrac{d}{b} = \dfrac{c}{a};\dfrac{ d}{c} = \dfrac{b}{a}\)
Solve problem 6.36 page 21 Math 7 Textbook Connecting knowledge volume 2
The inch (pronounced inches and abbreviated in) is the name of a unit of length in the American System of Measurements. Know that 1 in = 2.54 cm.
a) What is the height of a person 170 cm tall (round to the nearest unit)?
b) Is a person’s height in centimeters proportional to his or her height in inches? If yes, what is the ratio?
Solution method
Length (in cm) = 2.54. Length (in inches)
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Detailed explanation
a) His height is:
170 : 2.54 \( \approx \)66.9 \( \approx \)67 ( inches)
b) A person’s height in centimeters is proportional to that person’s height in inches because they are related by the formula: Length (in centimeters) = 2.54. Length (in inches)
The scaling factor is 2.54.
Solution 6.37 page 21 Math 7 textbook Connecting knowledge volume 2
The measure of the three angles \(\widehat A,\widehat B,\widehat C\) of a triangle ABC is proportional to 5,6;7. Calculate the measure of the three angles of the triangle.
Solution method
The sum of the 3 angles of a triangle is 180 degrees.
Apply the property of the series of equal ratios:\(\dfrac{a}{b} = \dfrac{c}{d} = \dfrac{e}{f} = \dfrac{{a + c + e} }{{b + d + f}}\)
Detailed explanation
In triangle ABC there are: \(\widehat A + \widehat B + \widehat C = 180^\circ \)
Since the measure of the three angles \(\widehat A,\widehat B,\widehat C\) of triangle ABC is proportional to 5;6;7 so \(\dfrac{{\widehat A}}{5} = \dfrac {{\widehat B}}{6} = \dfrac{{\widehat C}}{7}\)
Applying the property of the series of equal ratios, we have:
\(\begin{array}{l}\dfrac{{\widehat A}}{5} = \dfrac{{\widehat B}}{6} = \dfrac{{\widehat C}}{7} = \ dfrac{{\widehat A + \widehat B + \widehat C}}{{5 + 6 + 7}} = \dfrac{{180^\circ }}{{18}} = 10^\circ \\ \Rightarrow \widehat A = 10^\circ .5 = 50^\circ \\\widehat B = 10^\circ .6 = 60^\circ \\\widehat C = 10^\circ .7 = 70^\circ \ end{array}\)
So the measures of the three angles \(\widehat A,\widehat B,\widehat C\) are \(50^\circ ;60^\circ ;70^\circ \)
Solve lesson 6.38 on page 21 Math 7 Textbook Connecting knowledge volume 2
Three teams of road workers were assigned three equal amounts of work. The first team completed the work in 4 days, the second team in 5 days and the third team in 6 days. Calculate the number of workers for each team knowing that the first team is 3 more than the second and that the workers’ productivity is the same throughout the working period.
Solution method
Let the number of workers in each team be x,y,z (people) respectively (x,y,z \( \in \)N*).
Number of workers and completion time are inversely proportional
Use the property of the series of equal ratios: \(\dfrac{a}{b} = \dfrac{c}{d} = \dfrac{e}{f} = \dfrac{{a – c}}{ {b – d}}\)
Detailed explanation
Let the number of workers in each team be x,y,z (people) respectively (x,y,z \( \in \)N*).
Since the number of workers in the first team is 3 more than the number of workers in the second team, x – y = 3
Since the workload is the same and the productivity of the machines is the same, the number of workers and the completion time are inversely proportional.
Applying the property of two quantities that are inversely proportional, we have:
4x=5y=6z
\(\begin{array}{l} \Rightarrow \dfrac{x}{{\dfrac{1}{4}}} = \dfrac{y}{{\dfrac{1}{5}}} = \dfrac {z}{{\dfrac{1}{6}}} = \dfrac{{x – y}}{{\dfrac{1}{4} – \dfrac{1}{5}}} = \dfrac{ 3}{{\dfrac{1}{{20}}}} = 3:\dfrac{1}{{20}} = 3.20 = 60\\ \Rightarrow x = 60.\dfrac{1}{4} = 15\\y = 60.\dfrac{1}{5} = 12\\z = 60.\dfrac{1}{6} = 10\end{array}\)
So 3 teams have 15 respectively; 12 and 10 workers.
