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

The importance of 'same' in Fraction questions

When it comes to fraction questions, pay attention to the 'same' word if it appears in the question. It will be the key to solve the question. Here is an example.

Question:

Adam and Bella had $5100 altogether. After Adam spent 4/5 of his money and Bella spent 5/7 of her money, they had the same amount of money left. How much money did each of them have at first?
 
Solution:

Key info: 'same amount of money left' 

Adam left 1 unit and Bella left 2 units. It means Adam's 1 unit is equal to Bella's 2 units. 

Change Adam's units to Bella's Units as shown in the table below:

17 units -> $5100
1 unit -> $300
Adam 10 units -> $3000
Bella 7 units -> $2100 

Number of Coins

Another question today.

Adam has 46 10-cent, 20-cent and 50-cent coins which add up to $14.60. There were 9 10-cent coins and the rest were 20-cent coins and 50-cent coins. How many 20-cent and 50-cent coins were there?

Solution:

10-cent coins --> $0.90
so 37 coins (20-cent coins + 50-cent coins) --> $14.60 - $0.90 = $13.70

Here we use assumption method to find the answer:

Assume all 37 coins are 50-cent coin --> 37 x $0.50 = $18.50
Difference --> $18.50 - $13.70 = $4.80
Exchange --> $0.50 - $0.20 = $0.30 
Number of 20-cent coin --> $4.80 ÷ $0.30 = 16
Number of 50-cent coin --> 37 - 16 = 21

Cost, Price, Discount and Profit

Here is a question from an anonymous visitor.

At Mrs Yu's shop, there were two vases for sale at $630 each. She sold one of them at this price and earned 40% of what she paid for it. She sold the other vase later at 20% discount. If the two vases had the same cost price, how much did Mrs Yu earn altogether?

Solution:

Vase 1: 140% --> $630, so 100% --> $450 (cost price), profit --> $630-$450= $180

Vase 2: 100% --> $630, so selling price is 80% --> $504,  profit --> $504 - $450= $54

Thus, total profit is $180+$54= $234

Note:

The common mistake in this question is the misunderstanding of " earned 40% of what she paid for it ". Students often use $630 x 40% to get the earning of first vase. However, this sentence actually means "what she paid for it " (cost price) is 100%. Therefore, the selling price $630 is 140%. 

Importance of 100%

Here is another question that my students always have problem with.

Allan had 40% more money than Ben. Charles had 20% more money than Allan. When Charles spend $1680, his amount of money decreased by 50%. Allan and Ben each spent 20% of their money. How much money did three of them left altogether?

The key for this question is to let Ben be 100% and be careful with Charles' percentage.

        84% --> $1680
        1% --> $20
        Total 276% --> $5520

The common mistake is the highlighted part - finding the percentage for Charles. 

Students normally just add 20% to Allan's 140% to find Charles. But this is wrong because Allan is not 100%.

**Note: We can only add the percentage directly when the base is 100% 

Importance of reading model

My student asked me this question:

If Alex uses his money to buy 8 similar notebooks, he will still have $6 more than Bella. If Bella uses some of her money to buy the same 8 notebooks, she will have $70 less than Alex. How much is the cost of 1 notebook?

This question can be solved easily using model.

16 notebooks + $6 --> $70

16 notebooks --> $64

1 notebook --> $4

Problem sums, even complex sums, can be solved easily using model drawings.  This is so as model drawing simplifies the problem and the pictorial representation helps pupils interpret the sum in a more comprehensible manner.   

However, the use of model drawing is just one of the many other heuristic strategies used in problem solving.  We will be sharing more effective methods, taught in Singapore Schools and we improvise them, to help you better guide your child in their Math homework.