Tag Archives: Math short cuts

Converting Fahrenheit to Celsius

To convert Fahrenheit to Celsius, we use the following formula,

                C = 5/9 (F – 32) = 10/18 (F – 32)

h

The trick here is to recognize that 5/9 is the same as 10/18.  Hence after deducting 32, we add a zero (multiply by 10) to the result and successively dividing by 2 and 9 (MSC 11 – Dividing by 9) to effectively divide by 18.

  • Step 1.  Subtract 32
  • Step 2.  Multiply by 10
  • Step 3. Divide by 2
  • Step 4. Divide by 9

Let us solve look at some examples.

Example 1:          100 oF = 37.8 oC

  • Step 1.  Subtract 32:         100 – 32 = 68
  • Step 2.  Multiply by 10:    68 x 10 = 680
  • Step 3.  Divide by 2:         680 ÷ 2 = 340
  • Step 4.  Divide by 9:         340 ÷ 9 = 37 r. 7  =  37.7…  =  37.8

Example 2:           145oF = 62.8oC

  • Step 1.  Subtract 32:          145 – 32 = 113
  • Step 2.  Multiply by 10:      113 x 10 = 1130
  • Step 3.  Divide by 2:           1130 ÷ 2 = 565
  • Step 4.  Divide by 9:           565 ÷ 9 = 62 r. 7  =  62.7…  =  62.8

Easy and simple!

 

MSC 9 – Dividing by 5, 50, 0.5, etc.

Five (5) is  ten divided by two ( 10 / 2 ), so to divide a number by 10/2, we

  • multiply it by 2, then
  • divide by 10.

Since dividing by 10 only involves moving the decimal point one place to the left, we can easily divide by 5 by just using doubling or multiplication by 2.

To divide a number by 5 we can either

Method A:

  • double the number first then
  • move the decimal point one place to the left.

Method B

  • move the decimal point first then
  • double the number.

We recommend the method B.

Let us try method A first.

Example 9.1: Find 164 ÷ 5

  • Double 16 is 32 and double is 8. So 164 x 2 is 328.
  • Move the decimal point one place to the left to make it 32.8.

Now let us use the method B.

Example 9.2: Find 832 ÷ 5

  • Shifting the decimal point of the multiplicand one place to the left will make it 83.2. This also fixed the decimal point for the answer.
  • Doubling it gives 166.4.

Example 9.3: Find 1348 ÷ 50

  • Since 50 is half of 100 or 102, we move the decimal point in 1348 two places to the left making it 13.48.
  • We then double it to make it 26.96.

Example 9.4:  Find 24.5 ÷ 0.5

  • 0.5 is half of 1 or 10so we do not have to adjust the decimal point. Just double the number to produce 49.0.
  • The answer becomes obvious if we double both the dividend and the divisor: 24.5 ÷ 0.5 = 49.0 ÷ 1 = 49

 Example 9.5: How many mint candies costing 50 centavos each can I buy with P 24.50?

 The figures here are the same as in Example 9.4 and the solution here clarifies the technique we used earlier: We can buy 2 candies for one peso; so for 24.50 pesos we can buy 24.5 x 2 or 49 candies.

Example 9.6: Find 376 ÷ 0.05

  • 0.05 is half of 0.1 or 10-1so we have to move the decimal point one place to the right, meaning we have to add a zero making it 3,760.
  • Doubling it would result to 7,520.

Exercise 9: Compute the following:

  1. )     370 ÷ 5 =
  2. )     535 ÷ 5 =
  3. )     2,367 ÷ 5 =
  4. )     9,898 ÷ 5 =
  5. )     4,656 ÷ 50 =
  6. )     24,579 ÷ 50 =
  7. )     5,836 ÷ 500 =
  8. )     34,785 ÷ 500 =
  9. )     4,524 ÷ 0.5 =
  10. )    3,645 ÷ 0.05 =

Answers to all exercises are found in the answer key.

Discover the 25 Math Short Cuts ( 25 MSC )!

 

Converting Kilograms to Pounds and Pounds to Kilograms

A Kilogram is 2.204622622 pounds. But for practical purposes we can consider 1 kilo =  2.2 pound conversion. So we can convert kilos to pounds by multiplying by 2 then by 1.1or by 11 then by 0.2 depending on which is easier.

Kilos to pounds scale

Convert kilo to pounds and pounds to kilo

42 kilos is (42 x 2) x 1.1 = 84 x 1.1 = 92.4 lbs.

57 kilos is (57 x 2) x 1.1 = 114 x 1.1 = 125.4 lbs

Or (57 x 11) x .2 = 627 x .2 = 125.4 lbs

303 kilos is (303 x 11) x.2 = 3333 x .2 = 666.6 lbs.

A Pound is about 0.45359 kgs. We can use 1 pound = 0.45 kilograms which is about 99.2% accurate for ordinary computations. To multiply by 0.45 we can multiply by 0.9 then divide by 2 or vice versa.

35 lbs is (35 x 0.9) ÷  2 = 31.5  ÷ 2 = 15.75 kgs

78 lbs is ( 78 x .09) ÷  2 = 70.2  ÷  2= 35.1 kgs

120 lbs is ( 120 ÷ 2) x .9 = 60 x 0.9 = 54 kgs.

540 lbs is (540  ÷ 2) x 0.9 = 270 x 0.9 = 243kg

Note that if we multiply 0.45 by 2.2 we will get 0.99.

Simple, right?

MSC 8 – Multiplying by 9

Recall the x9 table.

9 x 1  = 09
9 x 2  = 18
9 x 3  = 27
9 x 4  = 36
9 x 5  = 45
9 x 6  = 54
9 x 7  = 63
9 x 8  = 72
9 x 9  = 81
9 x 10 = 90

If you look closely at the consecutive products of 9, you will see a pattern.The first digit of the product is in increasing order from 0 to 9; while the second digit is decreasing from 9 to 0.

There is also a pattern on how we arrive at the product.  When 9 is multiplied to a number, we see that the product is a 2-digit number and the first digit of this product is always one less than the number being multiplied by 9.The second digit of the product is the number which when added to the first digit would give a total of 9.

Another way of getting this table is to consider 9 as (10 -1). Thus


9 x 1 = (10 -1) x 1 = 10 – 1 or 09;
9 x 2 = (10 -1) x 2 = 20 – 2 or 18;
9 x 3 = (10 -1) x 3 = 30 – 3 or 27

and so on. This is the general method we will use to multiply by 9

Example 8.1: 23 x 9 = 207

23 x 9 = 23 x (10 – 1) = 230 – 23 = 207

Example 8.2: 357 x 9 = 3213

357 x 9 = 357 x (10 – 1) = 3570 – 357 = 3213

We can develop a further short-cut for this method.

Step 1) Place a bar separating the last digit from the other digits of the multiplicand.
Step 2) Add 1 to the left hand part
Step 3) Subtract the result from the number. The difference is the first part of the answer.
Step 4) The second part is the ten’s complement of the last digit of the multiplicand (which simply means we subtract the last digit from 10)

Example 8.3: 23 x 9 =

Step 1)  Place a bar separating the last digit 3 from the other digits of the multiplicand, so we place a bar between 2 and 3

2 | 3

Step 2) Add 1 to the left hand part, so we add 1 to 2 to get 3.

2 + 1 = 3

Step 3)  Subtract the result from the number. The difference is the first part of the answer, so we subtract  3 from 23, to get the first part of the answer, 20

23 – 3 = 20
Step 4) The second part is the ten’s complement of the last digit of the multiplicand, which simply means we subtract the last digit 3 from 10 to get the second part 7.

10 – 3 = 7

The first part is 20, and the second part is 7.

2 | 3 x 9 = 20 | 7 or 207

 Example 8.4: 357 x 9 = 

Step 1) 35|7
Step 2) 35 + 1 = 36
Step 3)357 – 36 = 321
Step 4) 10 – 7 = 3

35 | 7 x 9 = 321|3 or 3213

Since the steps are very simple we can combine some of them.

Example 8.5: 1248 x 9 =

Step 1)  Subtract 125 ( or 124 + 1 ) from 1248 to get the first part.

1248 – 125 = 1123
Step 2) Subtract 8 from 10 to get the second part

10 – 8 = 2

Combine the first part 1123 and the second part 2

1248 x 9 = 1123|2

If the multiplicand is long or complicated we can always resort to written subtraction, which is simpler than multiplication.

Example 8.6: 35,784 x 9 =

(34,784 – 3,479 ) | (10 – 4 )

32,205 | 6

35,784 x 9 = 322,056

Example 8.7: Dealers are given 10% discounts of the selling prices. What is the dealer’s price for an item worth P 1150?

Solution: When you are given a 10% discount, you pay 90% of the cost.

1150 – 115 = P1,035

Here’s how to use your fingers in multiplying by 9:

Spread out your hands and represent each of your fingers with the numbers from 1 to 10 as shown.

finger math1

When multiplying a number by 9, say 9 x 4, simply bend the finger that represents 4.  Count the number of fingers on the left of that finger – this will serve as the first digit of your product and the number of fingers on the right side of your bent finger represents the second digit of your answer.

finger math 2

 Exercise 8; Use (10-1) in multiplying by 9

  1. )      6 x 9 =
  2. )      9 x 9 =
  3. )      35 x9 =
  4. )      49 x 9 =
  5. )      82 x 9 =
  6. )      148 x 9 =
  7. )      285 x 9 =
  8. )      0. 9 x 68 =
  9. )      90% of 675 =
  10. )  45% of 740 =

Answers to all exercises are found in the answer key.

Discover the 25 Math Short Cuts ( 25 MSC )!

 

Computing Commissions

by Ike Prudente

Before we started MATH-Inic, I planned to organize mental Math classes for parents so that they can teach their children at home. To promote my courses I thought of inviting them to free seminars, where I showed them examples of mental calculations.

A good friend, Past President Rosanna Panque of the Kiwanis Club of Siete Lagos who is a Unit Manager at Insular Life asked me to give an ice breaker at one of meetings of her associates. After discussing some short cuts, I asked my audience what they usually compute. As expected, they told me that they often want to know their commissions as soon as their clients bought a policy.

I asked them for an example which we can compute. They told me about a popular policy where the agent gets 45% of the first premium:

VYP: Ok, give me an amount of first premium.

Agent 1: P18,000

VYP: Can you compute  45% of P18,000 ?

Audience: (Silent)

VYP: What is P18,000 times 0.45?

Audience: (Silent)

VYP:  OK. Let us first divide P18,000 by 2. What do we get?

Audience: P9,000

VYP: Then we double 0.45. What do we get?

Audience: 0.90

VYP: So now we changed the problem to .90 x 9,000. Now, can you mentally compute that?

Audience: (Silent)

VYP: If we multiply 0.90 by 10 we get 9 and when we divide 9,000 by 10 we get 900. So our problem now becomes 9 x 900. Now can you compute that?

Agent 1: Yes! 9 x 9 is 81, so that is P8,100.

VYP: Correct! So you see, you can use doubling and halving to make calculations easier. Let us have another example. Give me another amount.

Agent 2: P 25,000!

I quickly realized that halving P25,000 would give P12,500 which has more non zero digits than the original number. So a different solution is needed.

VYP: Can you compute 45% of P25,000 ?

Audience: (Silent)

VYP: Let us try another solution. What is 10% of P25,000?

Audience: P2,500.

VYP: Deduct P2,500 from P25,000. What do we get?

Audience: P22,500.

VYP: Correct! Now since we deducted 10%, what remains is 90%. Now half 90% is 45%. What is half of P22,500?

Audience: P 11,250!

VYP: Do you have any other questions?

Agents: When will your Math-Inic classes start?

How much is the tuition fee?

Are you offering Math-Inic in Tiaong?

VYP’s Thoughts: Yes!Ayos! Panalo!

 

Note: Shortly after that, I realized that there are other shortcuts to computing 45% of P25,000.

1)      Halving and doubling together(MSC 7)

45% of P25,000 =  25% of P45,000

= .25 x 45,000

= .50 x 22,500

= 1 x 11,250

2)      Multiplying numbers ending in 5

45% of P25,000 = .45 x 25,000

= 45 x 250

Momentarily disregarding the ending zero in 250, we have

45 x 25 = (4×2)+(4+2)/2| 5 x 5

= 8+3|25

= 1125

We then attach the zero to give a final answer of 11,250. Further examples of this type of calculations will be given in MSC 23.

Doubling and Halving When Buying T-Shirts

By Ike Prudente

During the Philippine Science High School’s ANAC I (A National Alumni Convention I) where I represented Pisay Dos (Batch ’70)  I saw a booth selling Pisay T-Shirts. I was attracted to a nice red colored shirt. So I thought of buying one for myself. But then, I thought of my 5 children who are all Pisay alumni, Rose (class of 1990), Toto (class of 1992), Tata (class of 1995), Lelay (class of 1997) and Nica (class of 1999). I selected 5 more to fit their sizes. Oh, I almost forgot Toto’s wife Rheea, who is his batchmate. And so I chose another shirt for her.

When I asked the ladies manning the booth (who are both Pisay alumni) the cost of a shirt, they told me that it is P350.

“Wow, mahal pala! Baka wala akong dalang perang pambayad” I thought.

As one of the ladies took her calculator, I blurted out “4900 divided by two is 2450.” They were amazed that when she punched in 350 x 7 in her calculator, she got 2450.

“How did you do that?” she asked me.

“I doubled P350 to get  7 hundred which when you multiply by 7 will give you 49 hundred. Now dividing this by two will give you P2450.”

By mastering doubling and halving, I expanded my multiplication tables.

VYPrudente family