Mental Math Practice — 120 lessons
Explore the tool and practise with 120 lessons in four difficulty levels. Choose a topic, check your answer and take a level test. Your answers stay on this device.
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What this page does not do: predict future lottery outcomes.
Course: 120 lessons
- 1. Multiplication by 11
- 2. Multiplication by 25
- 3. Multiplication by 99
- 4. Division by 8
- 5. Multiplication by 5
- 6. Multiplication by 9
- 7. Multiplication by 12
- 8. Multiplication by 15
- 9. Multiplication by 19
- 10. Multiplication by 21
- 11. Multiplication by 50
- 12. Division by 5
- 13. Division by 25
- 14. Complement to 1000
- 15. Percentage as a fraction
- 16. To the next ten
- 17. To the next hundred
- 18. Carrying ones
- 19. Splitting a sum
- 20. Three addends
- 21. Subtract nine
- 22. Subtract ninety-nine
- 23. Distance to a hundred
- 24. Borrowing a ten
- 25. Two changes
- 26. Times six
- 27. Times seven
- 28. Times eight
- 29. Times thirteen
- 30. Two-digit factor
- 31. Squaring numbers ending in 5
- 32. Products below 100
- 33. Products above 100
- 34. On opposite sides of 100
- 35. Square of a sum
- 36. Crosswise multiplication
- 37. Multiplication by 125
- 38. Multiplication by 101
- 39. Multiplication by 999
- 40. Division by 125
- 41. Halving
- 42. Quartering
- 43. One eighth
- 44. Divide by seven
- 45. Checking a quotient
- 46. Ten percent
- 47. Five percent
- 48. Twenty percent
- 49. Three quarters
- 50. One percent
- 51. Square above twenty
- 52. Square below twenty
- 53. Square above fifty
- 54. Square above two hundred
- 55. Difference of consecutive squares
- 56. Base twenty
- 57. Base thirty
- 58. Base one hundred
- 59. Common midpoint
- 60. Product versus square
- 61. Difference of squares
- 62. Square below a round number
- 63. Sum of consecutive integers
- 64. 12.5 percent
- 65. Common factor
- 66. Multiply by 0.1
- 67. Divide by 0.2
- 68. Multiply by 0.25
- 69. Divide by 0.5
- 70. Decimal quotient
- 71. Ten percent increase
- 72. Twenty percent decrease
- 73. Increase then decrease
- 74. Two increases
- 75. Price before discount
- 76. Sum from one
- 77. First odd numbers
- 78. First even numbers
- 79. Five consecutive integers
- 80. Sum of multiples
- 81. Divide before multiply
- 82. Common factor in quotient
- 83. Half a product
- 84. Sum of multiples divided
- 85. Cancelling a fraction
- 86. Two negative factors
- 87. Negative product
- 88. Subtracting a negative
- 89. Square versus leading minus
- 90. Square of a negative
- 91. Unit price
- 92. Discounted price
- 93. Change
- 94. Two baskets
- 95. Unit price difference
- 96. Hours to minutes
- 97. Quarter hours to minutes
- 98. Unit pace
- 99. Minutes to seconds
- 100. Distance from time
- 101. Round-product correction
- 102. Rounding excess
- 103. Absolute approximation error
- 104. Relative error percent
- 105. Order-of-magnitude check
- 106. Missing addend
- 107. Missing factor
- 108. Base before a percentage
- 109. Before doubling
- 110. Mean and sum
- 111. Distributivity with subtraction
- 112. Doubling and halving
- 113. Regrouping factors
- 114. Factoring out
- 115. Product around a base
- 116. Last digit of a product
- 117. Remainder modulo nine
- 118. Verification by difference
- 119. Checking a quotient
- 120. Order of operations
Enter a fraction such as 1/3 or round as instructed beside the question (three or eight decimal places).
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1. Multiplication by 11
Level 1
Split 11 into 10 + 1. Multiply by 10 and add the original. This works even when digit sums need a carry.
Worked example
47 × 11 = 470 + 47 = 517.
Watch out
The digit-insertion shortcut needs a carry when the digit sum exceeds 9.
2. Multiplication by 25
Level 1
Since 25 = 100/4, multiply by 100 and divide by 4. For multiples of 4, dividing first is convenient.
Worked example
28 × 25 = 2800/4 = 700.
Watch out
Do not round intermediate division when the number is not divisible by 4.
3. Multiplication by 99
Level 1
99 = 100 − 1, so n × 99 = 100n − n. This is the distributive law.
Worked example
42 × 99 = 4200 − 42 = 4158.
Watch out
Subtract the whole original number, not just its last digit.
4. Division by 8
Level 1
Multiplying by 0.125 is dividing by 8. Halve three times. Choose a method suited to the numbers.
Worked example
248/8: 248 → 124 → 62 → 31.
Watch out
Do not confuse 0.125 with 1.25. This trains arithmetic, not lottery prediction.
5. Multiplication by 5
Level 1
Multiply by 10 and divide by 2.
Worked example
36 × 5 = 360/2 = 180.
Watch out
For odd numbers preserve the half before multiplying.
6. Multiplication by 9
Level 1
Nine times n is 10n−n.
Worked example
34 × 9 = 340−34=306.
Watch out
Subtract the whole number n.
7. Multiplication by 12
Level 1
Split 12 into 10+2.
Worked example
24 × 12 = 240+48=288.
Watch out
Add both products.
8. Multiplication by 15
Level 1
15n = 10n + 5n: ten times n plus half of that result.
Worked example
32 × 15 = 320+160=480.
Watch out
Take half of 10n, not n.
9. Multiplication by 19
Level 1
Use 20n−n.
Worked example
23 × 19 = 460−23=437.
Watch out
Keep the subtraction sign.
10. Multiplication by 21
Level 1
Use 20n+n.
Worked example
23 × 21 = 460+23=483.
Watch out
This adds, unlike the method for 19.
11. Multiplication by 50
Level 1
50n = 100n/2.
Worked example
46 × 50 = 4600/2=2300.
Watch out
Do not lose a zero when scaling.
12. Division by 5
Level 1
Double the number and divide by 10.
Worked example
135/5 = 270/10=27.
Watch out
Shift the decimal point after doubling.
13. Division by 25
Level 1
Multiply by 4 and divide by 100.
Worked example
375/25 = 1500/100=15.
Watch out
Do not divide by 4 instead of multiplying.
14. Complement to 1000
Level 1
Find the amount missing to 1000 by subtraction.
Worked example
1000−367=633; check: 367+633=1000.
Watch out
Check by addition.
15. Percentage as a fraction
Level 1
25% is 1/4, so divide by 4.
Worked example
25% of 84 is 21.
Watch out
Do not confuse 25% with multiplication by 25.
16. To the next ten
Level 1 · Addition by compensation
Round an addend, then subtract the excess.
Worked example
4 + 9 = ? 4+10−1 = 13
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
17. To the next hundred
Level 1 · Addition by compensation
Round an addend, then subtract the excess.
Worked example
40 + 99 = ? 40+100−1 = 139
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
18. Carrying ones
Level 1 · Addition by compensation
Round an addend, then subtract the excess.
Worked example
47 + 8 = ? 40+10+5 = 55
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
19. Splitting a sum
Level 1 · Addition by compensation
Round an addend, then subtract the excess.
Worked example
43 + 34 = ? (4+3)×10+7 = 77
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
20. Three addends
Level 1 · Addition by compensation
Round an addend, then subtract the excess.
Worked example
4 + 19 + 1 = ? 4+20 = 24
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
21. Subtract nine
Level 1 · Subtraction by compensation
Subtract a round number, then correct the difference.
Worked example
40 − 9 = ? 40−10+1 = 31
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
22. Subtract ninety-nine
Level 1 · Subtraction by compensation
Subtract a round number, then correct the difference.
Worked example
400 − 99 = ? 400−100+1 = 301
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
23. Distance to a hundred
Level 1 · Subtraction by compensation
Subtract a round number, then correct the difference.
Worked example
100 − 43 = ? 100−40−3 = 57
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
24. Borrowing a ten
Level 1 · Subtraction by compensation
Subtract a round number, then correct the difference.
Worked example
42 − 7 = ? 30+12−7 = 35
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
25. Two changes
Level 1 · Subtraction by compensation
Subtract a round number, then correct the difference.
Worked example
40 − 8 + 3 = ? 40−5 = 35
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
26. Times six
Level 1 · Distributive multiplication
Split one factor into convenient addends and add products.
Worked example
4 × 6 = ? 4×(5+1) = 24
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
27. Times seven
Level 1 · Distributive multiplication
Split one factor into convenient addends and add products.
Worked example
4 × 7 = ? 4×(5+2) = 28
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
28. Times eight
Level 1 · Distributive multiplication
Split one factor into convenient addends and add products.
Worked example
4 × 8 = ? 4×(10−2) = 32
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
29. Times thirteen
Level 1 · Distributive multiplication
Split one factor into convenient addends and add products.
Worked example
4 × 13 = ? 4×(10+3) = 52
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
30. Two-digit factor
Level 1 · Distributive multiplication
Split one factor into convenient addends and add products.
Worked example
43 × 3 = ? 40×3+3×3 = 129
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
31. Squaring numbers ending in 5
Level 2
For 10a + 5, the square is 100a(a+1) + 25. Multiply the part before 5 by its successor and append 25.
Worked example
35² = 3 × 4 × 100 + 25 = 1225.
Watch out
This form applies to integers ending in the digit 5.
32. Products below 100
Level 2
Write the factors as 100−a and 100−b. Their product is 100(100−a−b) + ab. The algebra handles carries correctly.
Worked example
97 × 96 = 100 × 93 + 12 = 9312.
Watch out
Do not simply concatenate deficits: ab ≥ 100 requires carrying hundreds.
33. Products above 100
Level 2
For 100+a and 100+b use 100(100+a+b) + ab. Surpluses replace the previous lesson’s deficits.
Worked example
104 × 107 = 11100 + 28 = 11128.
Watch out
With large surpluses, add numerically rather than concatenating digits.
34. On opposite sides of 100
Level 2
Expand (100−a)(100+b): 10000 + 100(b−a) − ab. The last term is negative.
Worked example
98 × 103 = 10000 + 100 − 6 = 10094.
Watch out
The sign of ab matters. Write the identity before doing it mentally.
35. Square of a sum
Level 2
Use (a+b)² = a² + 2ab + b². Split a number into tens and the remainder.
Worked example
23² = 400 + 120 + 9 = 529.
Watch out
Do not omit 2ab. The square of a sum is not the sum of squares.
36. Crosswise multiplication
Level 2
For (10a+b)(10c+d), use 100ac + 10(ad+bc) + bd. The middle term includes both cross-products.
Worked example
23 × 14 = 200 + 110 + 12 = 322.
Watch out
Preserve place value. Multiplying units can carry into tens.
37. Multiplication by 125
Level 2
125n = 1000n/8.
Worked example
24 × 125 = 24000/8=3000.
Watch out
Division by 8 can use three successive halvings.
38. Multiplication by 101
Level 2
101n = 100n+n.
Worked example
37 × 101 = 3700+37=3737.
Watch out
Digit concatenation has limits; the formula works generally.
39. Multiplication by 999
Level 2
999n = 1000n−n.
Worked example
42 × 999 = 42000−42=41958.
Watch out
Track borrowing across digit positions.
40. Division by 125
Level 2
Multiply by 8 and divide by 1000.
Worked example
1500/125 = 12000/1000=12.
Watch out
Track three decimal places of scaling.
41. Halving
Level 2 · Exact division
Split the dividend into multiples of the divisor. Check by multiplication.
Worked example
56 / 2 = ? 40/2+16/2 = 28
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
42. Quartering
Level 2 · Exact division
Split the dividend into multiples of the divisor. Check by multiplication.
Worked example
112 / 4 = ? 112/2/2 = 28
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
43. One eighth
Level 2 · Exact division
Split the dividend into multiples of the divisor. Check by multiplication.
Worked example
224 / 8 = ? 224/2/2/2 = 28
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
44. Divide by seven
Level 2 · Exact division
Split the dividend into multiples of the divisor. Check by multiplication.
Worked example
280 / 7 = ? 4×10 = 40
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
45. Checking a quotient
Level 2 · Exact division
Split the dividend into multiples of the divisor. Check by multiplication.
Worked example
12 / 4 = ? 4×3=12 = 3
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
46. Ten percent
Level 2 · Fractions and percentages
Percent means one hundredth. Recognise simple fractions before multiplying.
Worked example
10% × 80 = ? 80/10 = 8
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
47. Five percent
Level 2 · Fractions and percentages
Percent means one hundredth. Recognise simple fractions before multiplying.
Worked example
5% × 160 = ? 160/20 = 8
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
48. Twenty percent
Level 2 · Fractions and percentages
Percent means one hundredth. Recognise simple fractions before multiplying.
Worked example
20% × 60 = ? 60/5 = 12
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
49. Three quarters
Level 2 · Fractions and percentages
Percent means one hundredth. Recognise simple fractions before multiplying.
Worked example
75% × 32 = ? 32/4×3 = 24
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
50. One percent
Level 2 · Fractions and percentages
Percent means one hundredth. Recognise simple fractions before multiplying.
Worked example
1% × 1200 = ? 1200/100 = 12
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
51. Square above twenty
Level 2 · Squares around a base
(B±d)²=B²±2Bd+d². Do not omit the cross term.
Worked example
24² = ? 400+40×4+4² = 576
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
52. Square below twenty
Level 2 · Squares around a base
(B±d)²=B²±2Bd+d². Do not omit the cross term.
Worked example
16² = ? 400−40×4+4² = 256
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
53. Square above fifty
Level 2 · Squares around a base
(B±d)²=B²±2Bd+d². Do not omit the cross term.
Worked example
54² = ? 2500+100×4+4² = 2916
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
54. Square above two hundred
Level 2 · Squares around a base
(B±d)²=B²±2Bd+d². Do not omit the cross term.
Worked example
204² = ? 40000+400×4+4² = 41616
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
55. Difference of consecutive squares
Level 2 · Squares around a base
(B±d)²=B²±2Bd+d². Do not omit the cross term.
Worked example
5² − 4² = ? 2×4+1 = 9
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
56. Base twenty
Level 2 · Symmetric products
(B−d)(B+d)=B²−d². Use the midpoint of both factors.
Worked example
16 × 24 = ? 20²−4² = 384
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
57. Base thirty
Level 2 · Symmetric products
(B−d)(B+d)=B²−d². Use the midpoint of both factors.
Worked example
26 × 34 = ? 30²−4² = 884
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
58. Base one hundred
Level 2 · Symmetric products
(B−d)(B+d)=B²−d². Use the midpoint of both factors.
Worked example
96 × 104 = ? 100²−4² = 9984
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
59. Common midpoint
Level 2 · Symmetric products
(B−d)(B+d)=B²−d². Use the midpoint of both factors.
Worked example
39 × 41 = ? 40²−1 = 1599
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
60. Product versus square
Level 2 · Symmetric products
(B−d)(B+d)=B²−d². Use the midpoint of both factors.
Worked example
40² − 37 × 43 = ? 3² = 9
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
61. Difference of squares
Level 3
(a−b)(a+b)=a²−b²: cross terms cancel.
Worked example
48 × 52 = 50²−2²=2496.
Watch out
Both distances from the midpoint must match.
62. Square below a round number
Level 3
(a−b)²=a²−2ab+b².
Worked example
49²=(50−1)²=2500−100+1=2401.
Watch out
The last term is positive.
63. Sum of consecutive integers
Level 3
1+…+n=n(n+1)/2; pair first and last terms.
Worked example
1+…+10=10 × 11/2=55.
Watch out
This formula starts at 1.
64. 12.5 percent
Level 3
12.5% is 1/8, so divide by 8.
Worked example
12.5% of 240 is 30.
Watch out
Percent includes division by 100.
65. Common factor
Level 3
ax+bx=(a+b)x; factor out the shared multiplier.
Worked example
7 × 13 + 3 × 13 = 10 × 13 = 130.
Watch out
The factor must match in both products.
66. Multiply by 0.1
Level 3 · Scaling and decimals
Move the decimal according to powers of ten; simultaneous scaling can preserve a value.
Worked example
4 × 0.1 = ? 4/10 = 0.4
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
67. Divide by 0.2
Level 3 · Scaling and decimals
Move the decimal according to powers of ten; simultaneous scaling can preserve a value.
Worked example
4 / 0.2 = ? 4×5 = 20
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
68. Multiply by 0.25
Level 3 · Scaling and decimals
Move the decimal according to powers of ten; simultaneous scaling can preserve a value.
Worked example
32 × 0.25 = ? 32/4 = 8
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
69. Divide by 0.5
Level 3 · Scaling and decimals
Move the decimal according to powers of ten; simultaneous scaling can preserve a value.
Worked example
4 / 0.5 = ? 4×2 = 8
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
70. Decimal quotient
Level 3 · Scaling and decimals
Move the decimal according to powers of ten; simultaneous scaling can preserve a value.
Worked example
1.2 / 0.3 = ? 12/3 = 4
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
71. Ten percent increase
Level 3 · Compound percentages
Successive changes multiply factors rather than add percentages.
Worked example
400 × 1.1 = ? 400+40 = 440
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
72. Twenty percent decrease
Level 3 · Compound percentages
Successive changes multiply factors rather than add percentages.
Worked example
400 × 0.8 = ? 400−80 = 320
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
73. Increase then decrease
Level 3 · Compound percentages
Successive changes multiply factors rather than add percentages.
Worked example
100 × 1.1 × 0.9 = ? 110×.9 = 99
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
74. Two increases
Level 3 · Compound percentages
Successive changes multiply factors rather than add percentages.
Worked example
100 × 1.2 × 1.2 = ? 120×1.2 = 144
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
75. Price before discount
Level 3 · Compound percentages
Successive changes multiply factors rather than add percentages.
Worked example
320 / 0.8 = ? 320×5/4 = 400
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
76. Sum from one
Level 3 · Series sums
An arithmetic series sum is n times the mean of its first and last terms.
Worked example
1 + 2 + … + 10 = ? 10×11/2 = 55
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
77. First odd numbers
Level 3 · Series sums
An arithmetic series sum is n times the mean of its first and last terms.
Worked example
1 + 3 + … + 9 = ? 5² = 25
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
78. First even numbers
Level 3 · Series sums
An arithmetic series sum is n times the mean of its first and last terms.
Worked example
2 + 4 + … + 10 = ? 5×6 = 30
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
79. Five consecutive integers
Level 3 · Series sums
An arithmetic series sum is n times the mean of its first and last terms.
Worked example
4 + 5 + 6 + 7 + 8 = ? 5×6 = 30
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
80. Sum of multiples
Level 3 · Series sums
An arithmetic series sum is n times the mean of its first and last terms.
Worked example
4 + 8 + 12 + 16 = ? 4×(1+2+3+4) = 40
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
81. Divide before multiply
Level 3 · Cancelling factors
Cancel common numerator and denominator factors, never addends.
Worked example
16 × 25 = ? 4×100 = 400
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
82. Common factor in quotient
Level 3 · Cancelling factors
Cancel common numerator and denominator factors, never addends.
Worked example
(4 × 3) / 4 = ? 3 = 3
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
83. Half a product
Level 3 · Cancelling factors
Cancel common numerator and denominator factors, never addends.
Worked example
8 × 3 / 2 = ? 4×3 = 12
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
84. Sum of multiples divided
Level 3 · Cancelling factors
Cancel common numerator and denominator factors, never addends.
Worked example
(12 + 20) / 4 = ? 3+5 = 8
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
85. Cancelling a fraction
Level 3 · Cancelling factors
Cancel common numerator and denominator factors, never addends.
Worked example
12 / 30 = ? 4/10 = 0.4
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
86. Two negative factors
Level 3 · Signed arithmetic
A product of two negatives is positive. Parentheses determine order.
Worked example
(−4) × (−3) = ? 4×3 = 12
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
87. Negative product
Level 3 · Signed arithmetic
A product of two negatives is positive. Parentheses determine order.
Worked example
(−4) × 3 = ? −(4×3) = -12
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
88. Subtracting a negative
Level 3 · Signed arithmetic
A product of two negatives is positive. Parentheses determine order.
Worked example
4 − (−3) = ? 4+3 = 7
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
89. Square versus leading minus
Level 3 · Signed arithmetic
A product of two negatives is positive. Parentheses determine order.
Worked example
−(4²) = ? −16 = -16
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
90. Square of a negative
Level 3 · Signed arithmetic
A product of two negatives is positive. Parentheses determine order.
Worked example
(−4)² = ? 4×4 = 16
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
91. Unit price
Level 4 · Project: shopping
Convert prices to matching units first. These are exercises with fictional data.
Worked example
24 / 6 = ? total/quantity = 4
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
92. Discounted price
Level 4 · Project: shopping
Convert prices to matching units first. These are exercises with fictional data.
Worked example
400 × 0.75 = ? 400/4×3 = 300
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
93. Change
Level 4 · Project: shopping
Convert prices to matching units first. These are exercises with fictional data.
Worked example
100 − 36 = ? 100−(40−4) = 64
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
94. Two baskets
Level 4 · Project: shopping
Convert prices to matching units first. These are exercises with fictional data.
Worked example
4 × 12 + 3 × 12 = ? (4+3)×12 = 84
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
95. Unit price difference
Level 4 · Project: shopping
Convert prices to matching units first. These are exercises with fictional data.
Worked example
12 / 3 − 6 / 2 = ? 4−3 = 1
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
96. Hours to minutes
Level 4 · Project: time and rate
Convert units before dividing. Pace and speed are reciprocal ratios.
Worked example
4 × 60 = ? 4×6×10 = 240
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
97. Quarter hours to minutes
Level 4 · Project: time and rate
Convert units before dividing. Pace and speed are reciprocal ratios.
Worked example
4 × 15 = ? 4×(10+5) = 60
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
98. Unit pace
Level 4 · Project: time and rate
Convert units before dividing. Pace and speed are reciprocal ratios.
Worked example
24 / 6 = ? time/distance = 4
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
99. Minutes to seconds
Level 4 · Project: time and rate
Convert units before dividing. Pace and speed are reciprocal ratios.
Worked example
4.5 × 60 = ? 4×60+30 = 270
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
100. Distance from time
Level 4 · Project: time and rate
Convert units before dividing. Pace and speed are reciprocal ratios.
Worked example
60 × 4 / 2 = ? 60/2×4 = 120
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
101. Round-product correction
Level 4 · Project: estimation
Estimate the scale first, then calculate exactly. Estimation checks rather than replaces the result.
Worked example
4 × 98 = ? 4×100−8 = 392
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
102. Rounding excess
Level 4 · Project: estimation
Estimate the scale first, then calculate exactly. Estimation checks rather than replaces the result.
Worked example
4 × 100 − 4 × 99 = ? 4×(100−99) = 4
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
103. Absolute approximation error
Level 4 · Project: estimation
Estimate the scale first, then calculate exactly. Estimation checks rather than replaces the result.
Worked example
1000 − 997 = ? |1000−997| = 3
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
104. Relative error percent
Level 4 · Project: estimation
Estimate the scale first, then calculate exactly. Estimation checks rather than replaces the result.
Worked example
(102 − 100) / 100 × 100 = ? 2/100×100 = 2
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
105. Order-of-magnitude check
Level 4 · Project: estimation
Estimate the scale first, then calculate exactly. Estimation checks rather than replaces the result.
Worked example
400 × 30 = ? 4×3×1000 = 12000
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
106. Missing addend
Level 4 · Project: inverse problems
Recover a missing value with the inverse operation and check by substitution.
Worked example
(7) − 4 = ? x+4=7 = 3
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
107. Missing factor
Level 4 · Project: inverse problems
Recover a missing value with the inverse operation and check by substitution.
Worked example
12 / 4 = ? 4x=12 = 3
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
108. Base before a percentage
Level 4 · Project: inverse problems
Recover a missing value with the inverse operation and check by substitution.
Worked example
20 / 0.05 = ? .05x=20 = 400
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
109. Before doubling
Level 4 · Project: inverse problems
Recover a missing value with the inverse operation and check by substitution.
Worked example
(8 − 3) / 2 = ? 2x+3=8 = 2.5
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
110. Mean and sum
Level 4 · Project: inverse problems
Recover a missing value with the inverse operation and check by substitution.
Worked example
4 × 5 = ? sum=count×mean = 20
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
111. Distributivity with subtraction
Level 4 · Project: choosing a method
Choose a simplifying decomposition, then check algebraic equivalence.
Worked example
4 × 97 = ? 4×(100−3) = 388
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
112. Doubling and halving
Level 4 · Project: choosing a method
Choose a simplifying decomposition, then check algebraic equivalence.
Worked example
8 × 35 = ? 4×70 = 280
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
113. Regrouping factors
Level 4 · Project: choosing a method
Choose a simplifying decomposition, then check algebraic equivalence.
Worked example
25 × 4 × 4 = ? (25×4)×4 = 400
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
114. Factoring out
Level 4 · Project: choosing a method
Choose a simplifying decomposition, then check algebraic equivalence.
Worked example
4 × 17 + 4 × 3 = ? 4×20 = 80
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
115. Product around a base
Level 4 · Project: choosing a method
Choose a simplifying decomposition, then check algebraic equivalence.
Worked example
104 × 103 = ? 10000+100×7+12 = 10712
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
116. Last digit of a product
Level 4 · Project: error checking
Check using another method. A last digit or remainder can detect an error but does not prove the whole result correct.
Worked example
4 × 3 mod 10 = ? remainder after division by 10 = 2
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
117. Remainder modulo nine
Level 4 · Project: error checking
Check using another method. A last digit or remainder can detect an error but does not prove the whole result correct.
Worked example
43 mod 9 = ? digit sum mod 9 = 7
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
118. Verification by difference
Level 4 · Project: error checking
Check using another method. A last digit or remainder can detect an error but does not prove the whole result correct.
Worked example
48 − 40 = ? 4×(12−10) = 8
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
119. Checking a quotient
Level 4 · Project: error checking
Check using another method. A last digit or remainder can detect an error but does not prove the whole result correct.
Worked example
4 × 3 − 12 = ? product−dividend = 0
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
120. Order of operations
Level 4 · Project: error checking
Check using another method. A last digit or remainder can detect an error but does not prove the whole result correct.
Worked example
4 + 3 × 2 = ? 4+(3×2) = 10
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
Test: choose your scope
Choose a lesson range and test length. Topics are drawn without repetition, up to the number available in that range. Answers appear after the test; there is no time limit.
Further reading
Original exercises by Pewniaki. The mental-maths identities are explained algebraically; no claim of historical origin or guaranteed speed advantage is made.