"Twice more than (two times more)" vs "Twice as much"

A common mistake by my student is "twice more than", they often mistake it as "twice as much as". Actually they are different:

•   Twice as much = 2 X the base
•      Twice more than (Two times more) = 2 X the base + the base

Here is an example:

Tom's present age is twice more than Sam's age years ago. Today, Sam's age is Tom's age the same number of years ago. How old is Tom now if the sum of their present age is 30?

Solution:

In this question, "Tom's present age is twice more than Sam's age years ago", means Tom's present age is 3u, Sam's age years ago is 1u.

Same Denominator in Fraction Questions

I have received some feedback from one visitor regarding my previous post. She asked me to post an example of same denominator question. Here it is.

Question:

Adam and Ben had same amount of money. Adam spent 1/4 of his money and Ben spent 2/5 of his money. Adam spent $300 less than Ben. How much did the boys have altogether at first?

Solution:

Adam spent 1/4

Ben spent 2/5

Adam and Ben had same amount of money --> Adam's 4u has to be same as Ben's 5u

Adam 1/4 (multiply both numerator and denominator by 5) --> 5/20

Ben 2/5 (multiply both numerator and denominator by 4) --> 8/20

Adam spent 5u and Ben spent 8u

8u - 5u = 3u --> $300

1u --> $100

The boys had 16u altogether at first --> 16 x $100 = $1600

Same Numerator in Fraction Questions

Adam and Ben had $3400 altogether. After Adam spent 3/4 of his money and Ben spent 7/9 of his money, they had the same amount of money left. How much money did each boy have at first?

Solution:

Normally students tend to use model to solve this question. However, it is actually unnecessary to draw model. It can be solved just using the fraction itself.

Fraction of what Adam left --> 1/4

Fraction of what Ben left --> 2/9

They had the same amount of money left, so Adam's 1 unit needs to be changed to 2 units so as to be the same as Ben's units.

1/4 --> 2/8

At first Adam had 8 units, Ben had 9 units.
17u --> $3400
1u --> $200

Adam 8u --> $1600
Ben 9u --> $1800

Note:
The key for this kind of questions is to change either numerator or denominator to be the same depends on the questions. 

Elimination Method

A question sent in by Jared

Mrs Goh had a sum of money. If she bought 32 apples and 9 pears, she would have $10.40 left. If she bought 12 apples and 24 pears, she would have $14.40 left. Given that she had $30, how many apples could she buy?

Solution:

'Buy two get one free' Question

Question:

Mr Lee bought some files at $4 each and sold them at $10 each. The customers who bought 2 files from him were given 1 file for free. Yesterday, all his customers bought either one or two files. At the end of the day, he had given away 120 files and had earned $1230. Find the number of customers who bought only one file. 

Solution:

Ratio + Before After (ALWAYS)

Here is another question from Kiasuparents on Ratio AGAIN! It seems ratio is really a headache for many students. 

A, B and C shared 360 cards. If A gives 10% of her cards to B and in turn B gives 2/7 of his cards to C, the ratio of A's cards to B's cards to C's cards will be 1:3:4. How many cards did B have at first ?

Solution:

Ratio in the end -> A : B : C = 1 : 3 : 4
Total 8u -> 360
1u -> 45
in the end, A -> 45, B -> 135, C -> 180
B gave 2/7 to C, so left 5units
5u -> 135
1u -> 27
After A gave 10% to B, B has 7u -> 7x27 = 189

For A, left 90%, 90% -> 45, 10% -> 5 (A gave 5 cards to B)

No. of cards B had at first: 189 - 5 = 184

Note:

At first, A had 50, B had 184, C had 126

Another Ratio Question (total unchanged)

Here is another ratio question from Kiasuparents

The cost of a birthday present was shared between Benjamin and Christopher. At first, Benjamin paid 2/3 of what Christopher paid. when Benjamin paid $50 more, he ended up paying 4/5 of what Christopher paid. how much was the cost of the present?

Soultion:

Hi use total unchanged (cost of the present is constant)
First B:C = 2:3 (total 5u)
After B:C = 4:5 (total 9u)

Change total units to 45u
First B:C = 2:3 (x9) --> 18:27
After B:C = 4:5 (x5) --> 20:25

2u --> 50
1u --> 25
Cost of present 45u --> $1125