Positive Rational Number Operations — Challenge
Fractions, decimals and percents are three costumes for the same number. Choose whichever makes the arithmetic easiest.
What these problems cover
These questions use 4 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Factor — A number that divides evenly into another.
- Least common denominator — The smallest expression every denominator divides into.
- Mixed number — A whole number with a fraction, like 212.
- Improper fraction — One whose top is bigger than its bottom, like 52.
Problem 1
What is 23+14?
- 37
- 1112
- 56
- 27
- 712
Worked answer
1112
Fractions can only be added once they share a denominator, and the least common denominator of 3 and 4 is 12.
23 = 812 and 14 = 312, so the sum is 1112.
Adding the numerators and denominators separately would give 37, which is less than 12 and so cannot be right.
Problem 2
What is 34÷58?
- 1532
- 56
- 65
- 3215
- 85
Worked answer
65
Dividing by a fraction is multiplying by its reciprocal: 34÷58 = 34·85.
34·85 = 2420 = 65.
The answer is greater than 1, which makes sense: 58 is smaller than 34, so it fits into it more than once.
Problem 3
Write 0.35 as a fraction in lowest terms.
- 720
- 3510
- 13
- 35
- 725
Worked answer
720
Two decimal places means hundredths: 0.35 = 35100.
Both parts divide by 5, giving 720, which cannot be reduced further because 7 is prime and does not divide 20.
Problem 4
Express 20% of 30% of 50 as a decimal.
- 0.03
- 0.3
- 3
- 15
- 30
Worked answer
3
Work outwards from the number: 30% of 50 is 0.30· 50 = 15.
Then 20% of 15 is 0.20· 15 = 3.
Equivalently 0.20· 0.30· 50 = 3 in one line.
Problem 5
What is 12+13+16?
- 16
- 23
- 56
- 1
- 32
Worked answer
1
The least common denominator of 2, 3 and 6 is 6.
12 = 36, 13 = 26 and 16 stays as it is.
36+26+16 = 66 = 1.
Problem 6
What is 78-56?
- 124
- 17
- 112
- 524
- 1124
Worked answer
124
The least common denominator of 8 and 6 is 24.
78 = 2124 and 56 = 2024.
The difference is 124. The two fractions are close in size, so a very small answer is what to expect.
Problem 7
What is 35 of 45?
- 15
- 18
- 25
- 27
- 75
Worked answer
27
Taking a fraction of a number means multiplying: 35· 45.
Divide by the denominator first: 45÷ 5 = 9, then multiply by the numerator: 9· 3 = 27.
Problem 8
What is (23)2·98?
- 12
- 34
- 43
- 23
- 13
Worked answer
12
Square the fraction first, squaring the top and the bottom: (23)2 = 49.
Then 49·98 = 3672 = 12. The nines cancel, which is the point of the question.
Problem 9
Write 1.25 as a mixed number in lowest terms.
- 114
- 115
- 125
- 53
- 112
Worked answer
114
The whole part is 1 and the decimal part is 0.25 = 25100 = 14.
So 1.25 = 114, which is 54 as an improper fraction.
Problem 10
A batch of bread uses 34 of a cup of flour. How many whole batches can be made from 6 cups of flour?
- 4.5
- 6
- 7
- 8
- 9
Worked answer
8
The question asks how many times 34 fits into 6, which is a division: 6÷34.
Multiply by the reciprocal: 6·43 = 243 = 8 batches, with no flour left over.
Multiplying by 34 instead would give 4.5, which is fewer batches than the flour allows.