Problem 1 :
One-sixth of the students in a band play flute and another one-fourth of the students play piano. What fraction of the students in the band play either the flute or piano?
Problem 2 :
Adam walks regular in the morning. He walks two-third of a mile in the first 35 minutes and three fifth of a mile in the next 25 minutes. How many miles does he walk in 1 hour?
Problem 3 :
Jessica is a sales representative in a store. In January, she had a sales target of $20000. She completed one-fourth of the target in the first two weeks, one-third in the third week and one-fifth in the last week of January. Find the amount of sales done by Jessica in January.
Problem 4 :
Noah has 120 cookies. He ate two-third of them and gave one-fifth of the remianing cookies to his friend. How many cookies did Noah give his friend?
Problem 5 :
One-third of John's present age is equal to his age 10 years ago. Find the present age of John.
Problem 6 :
In a triangle, measure of the first angle is one half of measure the second angle and measure of the second hangle is two-third of measure the third third angle. Find the measures of three angles of the triangle.
Problem 7 :
James had some money. He spent one-third of the money for buying candies and one-fourth of the remaning amoney for buying cookies. After having bought candies and cookies, he has left $15. Find the amount money James had in the begining.
Problem 8 :
Three students Andrew, Peter and Thomas participated in a quiz competition. Andrew scored ⅓ of the points scored by Peter and Peter scored ⅕ of the points sored by Thomas. If they all together had scored 760 points, how many points did Andrew, Peter and Thomas score separately?
1. Answer :
Fraction of the students in the band play either the flute or piano :
= ⅙ + ¼
Least common multiple of the denominators (6, 4) is 12.
Make each denominator as 12 by multiplying the numerator and denominator of the first fraction by 2 and the second fraction by 3.
= ⁽¹ ˣ ²⁾⁄₍₆ ₓ ₂₎ + ⁽¹ ˣ ³⁾⁄₍₄ ₓ ₃₎
= ²⁄₁₂ + ³⁄₁₂
= ⁽² ⁺ ³⁾⁄₁₂
= ⁵⁄₁₂
2. Answer :
35 minute + 25 minutes = 60 minutes
35 minutes + 25 minutes = 1 hour
⅔ miles + ⅗ miles + = ?
Least common multiple of the denominators 3 and 5 is 15.
Make each denominator as 15 by multiplying the numerator and denominator of the first fraction by 5 and the second fraction by 3.
⅔ + ⅗ = ⁽² ˣ ⁵⁾⁄₍₃ ₓ ₅₎ + ⁽³ ˣ ³⁾⁄₍₅ ₓ ₃₎
= ¹⁰⁄₁₅ + ⁹⁄₁₅
= ⁽¹⁰ ⁺ ⁹⁾⁄₁₅
= ¹⁹⁄₁₅
= 1⁴⁄₁₅
Adam walks 1⁴⁄₁₅ miles in 1 hour.
3. Answer :
Amount of sales Jessica completed in January :
= (⅓ + ¼ + ⅕) x 9000
Least common multiple of the denominatoirs (3, 4, 5) is 60.
Make each denominator as 60 by multiplying the numerator and denominator of each fraction by an appropriate number.
= (²⁰⁄₆₀ + ¹⁵⁄₆₀ + ¹²⁄₆₀) x 9000
= ⁽²⁰ ⁺ ¹⁵ ⁺ ¹²⁾⁄₆₀ x 9000
= ⁴⁷⁄₆₀ x 9000
= 47 x 150
= $7050
4. Answer :
Numberb of cookies eaten by Noah :
= ⅔ x 120
= 2 x 40
= 80
Number of cookies remaning :
= 120 - 80
= 40
Number of cookies given to the friend :
= ⅕ x 40
= 8
5. Answer :
Let x be the present age of John.
Jhon's age ten years ago = x - 10
Given : One-third of John's present age is equal to his age 10 years ago.
(⅓)x = x - 10
Multiply both sides by 3.
x = 3(x - 10)
x = 3x - 30
Subtract 3x from both sides.
-2x = -30
Divide both sides by -2.
x = 15
The present age of John is 15 years.
6. Answer :
Let x be the measure of the third amngle.
Measure of the second angle :
= ⅔ of the third angle
= (⅔)x
= ²ˣ⁄₃
Measure of the first angle :
= ½ of the second angle
= (½)(²ˣ⁄₃)
= ˣ⁄₃
In a triangle, three angles add up to 180°.
first angle + second angle + third angle = 180°
ˣ⁄₃ + ²ˣ⁄₃ + x = 180°
Multiply both sides by 3 to get rid of the denominators.
3(ˣ⁄₃ + ²ˣ⁄₃ + x) = 3(180°)
3(ˣ⁄₃) + 3(²ˣ⁄₃) + 3(x) = 3(180°)
x + 2x + 3x = 540°
6x = 540°
x = 90°
ˣ⁄₃ = ⁹⁰⁄₃ = 30°
²ˣ⁄₃ = ⁽² ˣ ⁹⁰⁾⁄₃ = 60°
The three angles of the triangle are 30°, 60° and 90°.
7. Answer :
Let x be the amount of money that James had in the begining.
Amount of money spoent for candies :
= (⅓)x
Amount of money spent for cookies :
= (¼)(⅔)x
= (⅙)x
Given : After having bought candies and cookies, James has left $15.
x - (⅓)x - (⅙)x = 15
Least common multiple of the denominators (3, 6) is 6. Multiply both sides of the equation by 6 to get rid of the denominators 3 and 6.
6[x - (⅓)x - (⅙)x] = 6(15)
6(x) - 6(⅓)x - 6(⅙)x = 90
6x - 2x - x = 90
3x = 90
Divide both sides by 3.
x = 30
James had $30 in the begning.
8. Answer :
Let a, p and t be the points scored by Andrew, Peter and Thomas score separately.
p = (⅕)t = ᵗ⁄₅
a = (⅓)p
a = (⅓)(ᵗ⁄₅)
a = ᵗ⁄1₅
Given : Andrew, Peter and Thomas together scored 760 points.
a + p + t = 760
ᵗ⁄1₅ + ᵗ⁄₅ + t = 760
Least common multiple of the denominators 15 and 5 is 15.
Multiply both sides of the euation by 15 to get rid of the denominatos 15 and 5.
15(ᵗ⁄1₅ + ᵗ⁄₅ + t) = 15(760)
15(ᵗ⁄1₅) + 15(ᵗ⁄₅) + 15(t) = 11400
t + 3t + 15t = 11400
19t = 11400
t = 600
a = ᵗ⁄1₅ = ⁶⁰⁰⁄₁₅ = 40
p = ᵗ⁄₅ = ⁶⁰⁰⁄₅ = 120
Therefore, Andrew scored 40 points, Peeter scored 120 points and Thomas scored 600 points.
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