"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 

'Set' Question

Question: 



Solution:

It can be solved using 'set'

1 set --> 8 bags of sweets and 1 bag of chocolate
Money collected for 1 set --> 8 x 5.5 + 2 = 46
No. of sets = 3128 / 46 = 68

No. of sweets sold --> 68 x 8 x 4 = 2176
No. of chocolates sold --> 68 x 8 = 544

Another Model Question

Question: 

Alex, Ben and Caleb had some stamps. Alex had 90 stamps more than Ben, and Ben had 10 stamps more than Caleb. After Alex had given Ben 95 of his stamps and Caleb bought some stamps, Alex and Caleb had the same number of stamps while Ben had thrice as much as either of them.

a) How many stamps did Caleb have at first?
b) How many stamps did they have altogether at first?

Solution:

Working Backwards

A fraction question sent in by Mr Chiow

Solution:

In the end: Z = L à 174 

After L gave 2/5, she had 3 units left.

So 3u à 174

1u à 58

Z received 2u from L, so Z originally had 174 – 2u à 174 – 116 = 58

Before L gave 2/5, she had 5 units. 5u à 290

Z had 58 after he gave 1/3, 2u à 58, 1u à 29 (given to L)

L had 290 after she received 29

L at first à 290 – 29 = 261 

Another common question on Ratio and Age

Question:

The ratio of the age of Mdm Huda to her son's age is 5:1 now. In 7 years time, she would be thrice as old as her son. What is the total age of Mdm Huda and her son now?

Solution:

Before ratio --> H : S = 5 : 1 (difference 4u)
After ratio --> H : S = 3 : 1 (difference 2u)
Age difference between Mdm Huda and her son is unchanged after 7 years

So make the difference same
After ratio (x2) --> H : S = 6 : 2 (difference 4u now, same as before)
1u (6u-5u) --> 7 years
Age of Mdm Huda now is 5u --> 5x7=35
Age of her son now is 1u --> 7

Two Math Questions

Two questions from Belnanna:

Question 1:

In a school science fair, there were exhibits from class A , class B and class C. Altogether, 25 exhibits came from class B and class C. In a total of 16 exhibits were not from class C and a total of 15 exhibits were not from class B.
How many exhibits were there altogether?

Solution:

16 exhibits not from C --> A + B = 16
15 exhibits not from B --> A + C = 15
2A + B + C = 31
B + C = 25
2A --> 6
A --> 3
B --> 13
C --> 12
Total exhibits --> 3 + 13 + 12 = 28


Question 2:

The length of a rectangular pond is 14 m and its breadth is 7 m. Fatimah wants to build a running path around the pond. The width of the path is 1/2 m. what is the area of the running path?

Solution:

Units and parts

Question:

Class 5A and 5B have the same number of pupils. The ratio of the number of boys in 5A to the number of boys in 5B is 2 : 3. The number of girls in 5A to the number of girls in 5B is 7 : 4. 
(A) Find the ratio of the number of boys to the number of girls in 5A.
(B) If there are 15 fewer girls than boys in 5B, how many pupils are there in 5B?

Solution:

5A boys : 5B boys = 2u : 3u
5A girls : 5B girls = 7p : 4p

5A and 5B same no. of pupils --> 2u + 7p = 3u + 4p --> 1u = 3p
5A boys : 5B boys = 2u : 3u = 6p : 9p

(A) 5A boys : 5A girls = 6 : 7

(B) 5B boys : 5B girls = 9 : 4 
5parts --> 15 
1p --> 3
No. of pupils in 5B is 13p --> 13 x 3 = 39

Cross Method for Fraction / Ratio Questions

Here is a question from visitor Belnanna:

There were 3/5 as many children as adults on a bus. At the next stop, 3 children boarded and 2 adults alighted from the bus.Then, there were 5/6 as many children as adults on the bus. How many children were there on the bus at first.

Solution:

P4 Question (Model)

Thanks Irene for sending in this P4 question. 

Question:

Patricia bought 2 bicycles n 5 tricycles. She paid $140 more for 5 tricycles than for 2 bicycles. If 2 bicycles cost as much as 3 tricycles, find the total cost of 1 bicycle n 1 tricycle.

Solution:

2 bicycles = 3 tricycles

changes all bicycles to tricycles in the model as followings:

Ratio, Ratio and Ratio!!

A very common Ration question!

Solution:

A simple P4 Question using Model

Question: 

Rachel had $80 more than Mary. If Mary gave Rachel $10, Rachel would have thrice as much as money as Mary. How much did they have altogether at first?

Solution:

A P5 Question