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SBT Prize At the end of chapter 6 Math 7 SBT Horizon
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Solve problem 1 page 17 SBT Math 7 Creative horizon episode 2 – CTST
Find a, b, c knowing:
a) \(\frac{a}{2} = \frac{b}{1} = \frac{c}{3}\) and \(a + b + c = 48\).
b) \(\frac{a}{2} = \frac{b}{3};\,\frac{b}{2} = \frac{c}{3}\) and \(a + c = 26\).
Detailed instructions for solving Lesson 1
Solution method
Apply property 2 of the series of equal ratios:
\(\frac{a}{b} = \frac{c}{d} = \frac{e}{f} = \frac{{a + c + e}}{{b + d + f}} = \frac{{a – c + e}}{{b – d + f}}\) (with \(b + d + f \ne 0,\,b – d + f \ne 0\)).
Detailed explanation
a) Applying the property of the series of equal ratios, we have: \(\frac{a}{2} = \frac{b}{1} = \frac{c}{3} = \frac{{a + b + c}}{{2 + 1 + 3}} = \frac{{48}}{6} = 8\)
So \(\frac{a}{2} = 8 \Rightarrow a = 16\); \(\frac{b}{1} = 8 \Rightarrow b = 8\); \(\frac{c}{3} = 8 \Rightarrow c = 24\).
b) We have: \(\frac{a}{2} = \frac{b}{3} \Rightarrow \frac{a}{4} = \frac{b}{6};\,\frac{b }{2} = \frac{c}{3} \Rightarrow \frac{b}{6} = \frac{c}{9}\), resulting in \(\frac{a}{4} = \frac {b}{6} = \frac{c}{9}\)
Applying the property of the series of equal ratios, we have: \(\frac{a}{4} = \frac{b}{6} = \frac{c}{9} = \frac{{a + c} }{{4 + 9}} = \frac{{26}}{{13}} = 2\)
So \(\frac{a}{4} = 2 \Rightarrow a = 8\); \(\frac{b}{6} = 2 \Rightarrow b = 12\).
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— *****
Solve problem 2 page 17 SBT Math 7 Creative horizon episode 2 – CTST
Based on the corresponding table of values in each of the following cases, state whether the two quantities are inversely proportional to each other.
a)
|
a |
\(first\) |
\(2\) |
\(3\) |
\(4\) |
\(5\) |
|
b |
\(60\) |
\(30\) |
\(20\) |
\(15\) |
\(twelfth\) |
b)
|
m |
\( – 2\) |
\( – first\) |
\(first\) |
\(2\) |
\(3\) |
|
n |
\( – twelfth\) |
\( – 24\) |
\(24\) |
\(twelfth\) |
\(9\) |
Detailed instructions for solving Lesson 2
Solution method
Two quantities that are inversely proportional y are related to x by the formula \(y = \frac{a}{x}\), or \(xy = a\). We say y is inversely proportional to x by the factor a.
Detailed explanation
a) \(ab = 1.60 = 2.30 = 3.20 = 4.15 = 5.12 = 60\) so a and b are inversely proportional.
b) \(2.12 \ne 3.9\) so m and n are not inversely proportional.
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Solve problem 3 page 18 SBT Math 7 Creative horizon episode 2 – CTST
Replace the appropriate number with ? in the following table such that the two quantities x and y are inversely proportional.
|
x |
\(5\) |
\(?\) |
\(3\) |
\(2\) |
\( – 4\) |
\( – 5\) |
|
y |
\(?\) |
\( – twelfth\) |
\(?\) |
\(?\) |
\(?\) |
\(8\) |
Detailed instructions for solving Lesson 3
Solution method
Determine the inverse ratio. Use the inverse ratio to find the unknown quantity.
Detailed explanation
We have \(xy = – 5.8 = – 40\)
|
x |
\(5\) |
\(\frac{{10}}{3}\) |
\(3\) |
\(2\) |
\( – 4\) |
\( – 5\) |
|
y |
\( – 8\) |
\( – twelfth\) |
\( – \frac{{40}}{3}\) |
\( – 20\) |
\(ten\) |
\(8\) |
–>
— *****
Solve problem 4 page 18 SBT Math 7 Creative horizon episode 2 – CTST
a) Find three numbers \(x,\,y,\,z\) satisfying \(x:y:z = 1:2:2\) and \(x + y + z = 25\).
b) Find three numbers \(a,\,b,c\) satisfying \(a:b:c = 3:4:5\) and \(a + b – c = 100\).
Detailed instructions for solving Lesson 4
Solution method
Step 1: Apply the definition of the series of equal ratios
If \(a:b:c = d:e:f\) then \(\frac{a}{d} = \frac{b}{e} = \frac{c}{f}\)
Step 2: Apply property 2 of the series of equal ratios:
\(\frac{a}{b} = \frac{c}{d} = \frac{e}{f} = \frac{{a + c + e}}{{b + d + f}} = \frac{{a – c + e}}{{b – d + f}}\) (with \(b + d + f \ne 0,\,b – d + f \ne 0\)).
Detailed explanation
a) From \(x:y:z = 1:2:2\) we have \(\frac{x}{1} = \frac{y}{2} = \frac{z}{2}\)
Applying the property of the series of equal ratios, we have:
\(\frac{x}{1} = \frac{y}{2} = \frac{z}{2} = \frac{{x + y + z}}{{1 + 2 + 2}} = \frac{{25}}{5} = 5\)
So \(\frac{x}{1} = 5 \Rightarrow x = 5\); \(\frac{y}{2} = 5 \Rightarrow y = 10\); \(\frac{z}{2} = 5 \Rightarrow z = 10\)
So \(x = 5;\,y = 10;\,z = 10\).
b) From \(a:b:c = 3:4:5\) we have \(\frac{a}{3} = \frac{b}{4} = \frac{c}{5}\)
Applying the property of the series of equal ratios, we have:
\(\frac{a}{3} = \frac{b}{4} = \frac{c}{5} = \frac{{a + b – c}}{{3 + 4 – 5}} = \frac{{100}}{2} = 50\)
So \(\frac{a}{3} = 50 \Rightarrow a = 150\); \(\frac{b}{4} = 50 \Rightarrow b = 200\); \(\frac{c}{5} = 50 \Rightarrow c = 250\)
So \(a = 150;\,b = 200;\,c = 250\).
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— *****
Solve problem 5 page 18 SBT Math 7 Creative horizon episode 2 – CTST
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A farm with 4 harvesters (with the same yield) finished a field in 6 hours. If there are 6 such reapers, how long will it take to finish the field?
Detailed instructions for solving Lesson 5
Solution method
Job completion time and number of harvesters are inversely proportional, so it is necessary to find the inverse proportionality factor and calculate the remaining quantity.
Detailed explanation
6 reapers will finish harvesting the field in \(\frac{{24}}{6} = 4\) hours.
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— *****
Solve lesson 6 page 18 SBT Math 7 Creative horizon episode 2 – CTST
Cuc wants to cut a rectangle with an area of 100 cm2. Let d (cm) and r (cm) be the two dimensions of the rectangle. Write a formula that shows the relationship between two quantities d and r.
Detailed instructions for solving Lesson 6
Solution method
The area of a rectangle is equal to the product of its two dimensions.
Detailed explanation
We have \(dr = 100\) cm2 .
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— *****
Solve problem 7 page 18 SBT Math 7 Creative horizon episode 2 – CTST
Let a be proportional to b by the scaling factor m and b to be proportional to c by the scaling factor n.
a) Let’s calculate a against b, calculate b against c.
b) Let’s calculate a in terms of c.
Detailed instructions for solving Lesson 7
Solution method
If y is proportional to x by the scaling factor k then \(y = kx\).
Detailed explanation
a) We know that a is proportional to b by the scale factor m, so \(a = mb\);
b is proportional to c by the scaling factor n, so \(b = nc\)
b) Replace \(b = nc\) into \(a = mb\) we get \(a = mnc\).
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— *****
Solve problem 8 page 18 SBT Math 7 Creative horizon episode 2 – CTST
Class 7A has 4 students who finished cleaning the class in 2 hours. If there are 16, how long will it take you to clean the classroom? (knowing that you are equally productive).
Detailed instructions for solving Lesson 8
Solution method
The number of students participating in cleaning and the time it takes to complete are inversely proportional. Need to find the inverse proportionality coefficient, based on the inverse proportionality coefficient to find the unknown quantity.
Detailed explanation
The number of students participating in cleaning and the completion time are inversely proportional, so the inverse coefficient is \(4.2 = 8\).
16 you will clean the class in \(\frac{8}{{16}} = 0.5\) hours \( = 30\) minutes.
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— *****
Solve lesson 9 page 18 SBT Math 7 Creative horizon episode 2 – CTST
Hoa wants to divide 1 kg of sugar evenly into n bags. Let p(g) be the mass of sugar in each bag. Show that n and p are inversely proportional and calculate p over n.
Detailed instructions for solving Lesson 9
Solution method
Determine the relationship between two quantities using a formula that concludes that the two quantities are inversely proportional.
Detailed explanation
Change \(1kg = 1000g\).
We have \(np = 1000\)(g). So n and p are two quantities that are inversely proportional.
The mass of sugar in each bag is \(p = \frac{{1000}}{n}\) (g)
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— *****
Solve problem 10 page 18 SBT Math 7 Creative horizon episode 2 – CTST
Class 7C has two friends who finished cleaning the school garden in 3 hours. If there are 6, how long will it take you to clean the school garden? (Knowing that you are equally productive)
Detailed instructions for solving Lesson 10
Solution method
The number of students participating in weeding in the garden and the completion time are inversely proportional. Need to find the inverse proportionality coefficient, based on the inverse proportionality coefficient to find the unknown quantity.
Detailed explanation
The inverse coefficient is \(2.3 = 6\).
6 you will mow in \(\frac{6}{6} = 1\) hours.
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