# Mental math solutions

Mathematics is not about numbers, equations, computations, or algorithms: it is about understanding.

William Paul Thurston

# Solutions for mental math quiz

1. 48 + 39 = ?

- Most people will add 40 + 30;
- 70;
- then add 8 + 9;
- 17;
- then add 70 + 17.

*Or*

- A more efficient strategy would be to add 48 + 30;
- 78;
- then 78 + 9;
- 87.

*Or*

Take one from 48 and add to 39 to make 40;

then 47 + 40 = 87.

2. 113 – 98 = ?

- Add 2 to 98 to make an easy number of 100;
- subtract 113 - 100;
- 13;
- Then add the 2 back;
- 13 + 2 = 15.

*Or*

- Add 2 to both numbers and get
- 115 - 100;
- Then subtract and get 15.

3. 14 x 5 x 7 = ?

- Factor 14;
- 2 * 7 * 5 * 7;
- rearrange (commutative property);
- 2 * 5 * 7 * 7;
- multiply (2 * 5) and (7 * 7);
- 10 * 49;
- multiply;
- 490

*Or*

- Break apart 14 and multiply 5 * 7;
- (10 + 4) + 35;
- multiply;
- 350 + (4 * 35);
- Use double double to multiply by 4 (35 + 35 = 70; 70 + 70 = 140)
- 350 + 140

4. 4 x 249 = ?

- Add one to 249;
- 4 * 250;
- Subtract 4 * 1;
- 1000 - 4;
- 996.

5. 6(37 + 63) + 18 = ?

- Add 37 + 63;
- 6 * 100 + 18;
- multiply;
- 600 + 18
- 618.

6. .25 x 9 = ?

- Think of .25 as 1/4;
- and 9 as 8 + 1;
- (14 of 8) + (1/4 of 1);
- 2 + 1/4;
- 2 1/4.

*Or*

- 1/4 of 8 = 2.
- 1/4 of 10 is 2.5
- ∴ 1/4 of 9 = 2.25

7. 1/2 + 1/3 = ?

- Think of a clock
- 1/2 = 30 minutes;
- 1/3 = 20 minutes;
- ∴ either
- 30/60 + 20/60 = 50/60.
- or 3/6 + 2/6 = 5/6

8. 152 – 75 = ?

- Remove 2;
- 150 - 75 = 75;
- Add 2 back;
- 75 + 2;
- 77.

9. 85/39 is closest to what integer?

- 39 * 2 = 78;
- 78 + 39 = 100 + ___;
- If 85 is closer to 78 (2) than 39 (1) or one hundred something (3), then the answer is ...
- Two.

10. 12.03 + .4 + 2.36 = ?

- 12.03 + .4;
- 12.43 + 2.36;
- 14.79

11. If you buy items priced (tax included) at $1.99, $2.99 and $3.98, your change from a $10 bill would be?

- ($2 + $3 + $4) - .04;
- $9 - .04 but before subtracting the 4 cents.
- I think that it will be the change that will be needed with the dollar;
- ∴ the change would be $1.04.

*Additional comment*

The ability to think of the 4 cents as money that isn't owed and money that will be change is one of the landmarks that Piaget identified in child development. To expect children to think in this manner before second grade is problematic.

12. 90% of 160 = ?

- Ten percent of 160 = 16;
- 90% = 100% - 10%;
- ∴ 160 -16;
- 144.

13. To the nearest dollar, the sale price of a dress listed at $49.35 and sold at 25% off is?

- 50% = 1/2;
- 25% = 1/4;

- 1/2 of $50 = $25;
- 1/4 of $25 = $12.50;

- Round $49.35 to $50;
- Round $12.50 to $13;
- ∴ $50 - $13 = $37.

14. The sum of the first ten odd positive integers (1 + 3 + 5 + 7 + 9 … 17 + 19) is ?

- (1 + 19) = 20;
- (3 + 17) = 20;
- (5 + 15) = 20;
- (7 + 13) = 20;
- (9 + 11) = 20;
- that's five pairs of integers that have a sum of 20.
- ∴ 5 * 20;
- 100.

15. The area of a square region of perimeter 20 is?

- Square with perimeter of 20;
- Square has four equal sides;
- 4 sides * 5 units long = 20 units long;

- If a square is 5 units long in each side;
- then its areaa wold be;
- 5 units * 5 units;
- 25 square units.

16. The ratio of the area of a circular region of radius one to that of the circumscribed square region is closest to .5, .6, .7 , .8, or .9

A circular region with area of one and circumscribed square region

- If the radius = 1;
- then the length of each side of the square is 2;
- ∴ the area of the square is 4 square units.
- If the radius = 1;
- then the area of the circle is π * 1
^{2}; - π or 3.14 square units.
- The ratio would be 3.14 : 4;
- a little more than 3/4 or .75
- ∴ .8.

17. The average (arithmetic mean) of 89, 94, 85, 90 and 97 is?

- Try 90.
- If 90 is the mean, then the total above and below should be the same.
- There is 11 more above 90 and 6 below 90.
- Since there are five numbers and 11 is 5 more that means there would be one more for each 5 to raise the average to 91.

- If the average is 91, then the total above is 9 and the total below is 9
- 82 + 2 = 91;
- 85 + 6 = 91; total below 91 = 9
- 90 + 1 = 91;

- 94 - 3 = 91;
- 97 - 6 = 91; Total above 91 = 9

18. If 4/6 = 16/x, then x = ?

- If 16 is 4 * 4;
- then 4/6 * 4/4;
- 16/24
- ∴ X = 24

19. If 2x + 3 = 25, then x = ?

- If 2 * 10 = 20 and 20 + 3 = 23;
- then 10 is too low.
- Try 2 * 11 = 2 and 22 +3 = 25;
- ∴ X = 11

20. √75 is closest to which integer?

- If 82 = 64 and 92 = 81;
- 64 + 11 = 75;
- 81 - 6 = 75;
- ∴ 9 is closest to √75.

21. To the nearest dollar, a 15% tip on a restaurant bill of $79.87 is?

- 10% of #80.00 is $8.00;
- 1/2 of $8.00 = $4.00'
- $8.00 + $4.00 = $12.00