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. 

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 

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

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: