What number must be added in each term of the fraction 7 by 31 so that it may become 5 by 17?

What number should be added to each of these numbers 12, 22, 42 and 72 so that the resulting numbers may be in proportion?

Answer

Hint: Product of extremes is equal to the product of mean. Proportional numbers are represented as \[a:b::c:d\] , where \[a,d\] are the extremes and \[b,c\] are known as the mean. In this question four proportional numbers are given so first we will add a common unknown number to them and then we will use means and extremes property to find the unknown number.

Complete step-by-step answer:
Given the four proportional numbers \[12:22::42:72\]
Let \[x\] be the number added to each of these numbers, so the new numbers become
First number \[ = 12 + x\]
Second number \[ = 22 + x\]
Third number \[ = 42 + x\]
Fourth number \[ = 72 + x\]
Now after adding an unknown numbers, since the resulting numbers need to be in proportion so we can write these numbers as
 \[12 + x:22 + x::42 + x:72 + x\]
Now we know the means and extremes property of proportionality where the product of extremes is equal to the product of mean, hence we can write
 \[\left( {12 + x} \right) \times \left( {72 + x} \right) = \left( {22 + x} \right) \times \left( {42 + x} \right)\]
Now by solving this
 \[
\Rightarrow 12 \times 72 + 12x + 72x + {x^2} = 22 \times 42 + 22x + 42x + {x^2} \\
\Rightarrow 864 + 84x = 924 + 64x \\
\Rightarrow 84x - 64x = 924 - 864 \\
\Rightarrow 20x = 60 \\
\Rightarrow x = 3 \;
 \]
Therefore we get the value of \[x = 3\]
Hence we can say if we add 3 to the numbers 12, 22, 42 and 72 the resulting numbers will be in proportion.
The resulting numbers are \[15:25::45:75\]
So, the correct answer is “3”.

Note: The numbers are proportional when the ratio of the LHS of the proportions is equal to the RHS of the proportion. To check if the numbers are in proportion we just find their ratio of both the sides.
The given numbers \[12:22::42:72\] are not in proportion since their ratios are not equal
 \[\dfrac{{12}}{{22}} = \dfrac{{42}}{{72}}\]
Now if we add 3 to each numbers then they become proportional
 \[
  \dfrac{{15}}{{25}} = \dfrac{{45}}{{75}} \\
  \dfrac{3}{5} = \dfrac{3}{5} \\
 \]

This page contains all the common types of PSLE Math questions and how to solve them.

Make sure you Bookmark this page as I will be updating this page with more types of PSLE Math questions.

Before you forget, download our PSLE Math Mind-Maps for free (for a limited time only). Click the download button below to download. 

1. Remainder Concept (Branching)

The first type of common PSLE Math questions is on remainder concept. This is usually tested in Fractions and Percentage questions for PSLE Math. When you see the keyword “remainder”, I recommend you to use this method called “Branching” or “Model Drawing” method. You will learn more about branching in the example below. 

John spent 3/5 of his money on books and 1/3 of the remainder on a wallet.

If John had $16 left, how much did he have at first?

Fraction of Money Left = 2/3 × 2/5 = 4/15

4 units = $16

15 units = $16 ÷ 4 × 15 = $60 (Answer)

2. Equal Fractions Concept

I have seen this concept being tested so many times in PSLE Math and yet, so many students failed to identify or apply the concept correctly. In this concept, you basically need to make the numerator the same and compare the denominator. 

There are 836 students in a school. 7/10 of the boys and 7/8 of the girls take bus to school. The number of boys who do not take bus is twice the number of girls who do not take bus. How many girls do not take bus?

Step 1: Find the fractions of boys and girls who do not take bus

Boys –> 1 – 7/10 = 3/10

Girls –> 1 – 7/8 = 1/8

Step 2: Compare the boys and girls who do not take bus

 3/10 of boys = 2 × 1/8 of girls  (The number of boys who do not take bus is twice the number of girls who do not take bus)

3/10 of boys = 1/4 of girls

Step 3: Make the numerators the same

1/4 = 3/12

Step 4: Compare the denominators

Boys : Girls

= 10 : 12 = 5 : 6

Step 5: Find the total number of units and equate it to the total boys and girls

5u + 6u = 11u

11u = 836

Step 6: Find 1 unit

1u = 76

Step 7: Find the total number of girls

6u = 456 (Total girls)

Step 8: Find the girls who do not take bus

1/8 × 456 = 57 girls (Answer)

3. More Than, Less Than, As Many As

Model drawingis used extensively in Singapore Math. I recommend you to use this method when you see the word “more than”, “less than” or “as many as” in Whole Number or Fractions questions. Watch the video below to learn how to cut models to compare them easily. 

James had 120 more marbles than Dan. After James lost 1/5 of his marbles and Dan lost 3/4 of his marbles, James had 184 more marbles than Dan. How many marbles did Dan have at first?

Click the video solutions below.

4. Constant Part Concept

The next type of common PSLE Math questions is on Constant Part Concept which is commonly tested in Ratio questions. The word “Constant” means remain the same. You need to identify the part which remain constant and make them equal in both ratios. 

Ali and Billy have money in the ratio of 5 : 6. After Billy spent $16, the ratio became 3 : 2. How much money does Billy have in the end?

Step 1: Make the ratio for Ali the same

Before:

A : B

= 5 : 6

= 15 : 18

After:

A : B

= 3: 2

= 15 : 10

Step 2: Find the difference between Billy’s starting amount and ending amount

18u – 10u = 8u

Step 3: Find 1 unit

8u = $16

1u = $2

Step 4: Find the amount for Billy in the end

10u = $20 (Ans)

4. Constant Total Concept

Under questions which involve “Internal Transfer”, the total remains the same. You can apply this concept when you see this type of PSLE Math questions. 

Ali and Billy have money in the ratio of 5 : 4. After Ali gave Billy $20, they have an equal amount of money. How much money does Billy have in the end?

Step 1: Make the total for Ali and Billy to be the same

Before:

A : B : Total

= 5 : 4 : 9

= 10 : 8 : 18

After:

A : B : Total

= 1 : 1 : 2

= 9 : 9 : 18

Step 2: Find the difference between Ali’s starting amount and ending amount

10u – 9u = 1u

Step 3: Find 1 unit

1 unit = $20

Step 4: Find Billy’s amount in the end

9 units = $180 (Ans)

4. Constant Difference Concept

For questions relating to age, the age difference between 2 people will always remain the same. 

The ages of Ali and Billy are in the ratio of 4 : 7. In 3 years’ time, their ages will be in the ratio of 3 : 5. How old is Billy now?

Step 1: Make the difference for Ali and Billy the same

Before:

A : B : Difference

= 4 : 7 : 3

= 8 : 14 : 6

After:

A : B : Difference

= 3 : 5 : 2

= 9 : 15 : 6

Step 2: Find the difference between Ali’s starting age and final age

9u – 8u = 1u

Step 3: Find 1 unit

1 unit = 3 years

Step 4: Find Billy’s age now

14 units = 42 years old (Ans)

5. Everything Changed Concept (Units and Parts)

This is a more challenging type of PSLE Math questions. Both sides of the ratio changed by different amounts. I recommend “Units and Parts” to solve this type of questions. 

The ratio of Ali’s money to Billy’s money was 2 : 1. After Ali saved another $60 and Billy spent $150, the ratio became 4 : 1. How much money did Ali have at first?

Step 1: Write down the starting ratio and apply the changes.

  A : B

= 2u : 1u

+60 : – 150

2u + 60 : 1u – 150

Step 2: Compare the final units with the final ratio.

A              :     B

= 2u + 60 : 1u – 150

= 4                :      1

Step 3: Cross multiply the final units with the final ratio

1 × (2u + 60) = 4 × (1u – 150)

2u + 60 = 4u – 600

Step 4: Solve for 1 unit

4u – 2u = 600 + 60

2u = $660 (Ans)

6. Part-Whole Concept

This is another common type of PSLE Math questions which you need to be clear on the relationship between the “part” and the “whole”. 

Kelly spent 1/3 of her money on 5 pens and 11 erasers. The cost of each pen is 3 times the cost of each eraser. She bought some more pens with 3/4 of her remaining money. How many pens did she buy altogether?

Step 1: Write down the ratio of the cost of pen : eraser

P : E = 3u : 1u

Step 2: Find the fraction spent on the extra pens

1 – 1/3 = 2/3 (Remainder)

3/4 × 2/3 = 1/2 (Fraction spent on extra pens)

Step 3: Find the total cost of 5 pens and 11 erasers

5 × 3u + 11 × u = 26u (Total cost of 5 pens and 11 erasers)

Step 4: Find the total amount of money in terms of units

26u × 3 = 78u (Total amount of money)

Step 5: Find the total cost of the extra pens

1/2 × 78u = 39u (Total cost of extra pens)

Step 6: Find the number of extra pens

39u ÷ 3u = 13

Step 7: Find the total number of pens

13 + 5 = 18 pens (Answer)

7. Excess and Shortage Concept

My students always tell me, “Mr Jimmy, this type of questions again!! It always come out in my exams and I always don’t know how to do!” 

Thank goodness you have stumbled onto this website!! Learn the excess and shortage method below to tackle this type of questions. 

Tom packed 5 balls into each bag and found that he had 8 balls left over. If he packed 7 balls into each bag, he would need another 4 more balls.
a) How many bags did he have?

b) How many balls did he have altogether?

Step 1: Find the difference in the number of balls in each bag

7 – 5 = 2

Step 2: Find the total difference of the balls in the bags of 5 and bags of 8

8 + 4 = 12

Step 3: Find the total number of bags

12 ÷ 2 = 6 bags (Ans for a)

Step 5: Find the total number of balls

6 × 5 + 8 = 38

Or

6 × 7 – 4 = 38 balls (Ans for b)

7. Gap and Difference Concept

This type of PSLE Math questions requires you to find the difference and use it to solve the question. 

Ali and John went on a trip together with the same amount of money. Ali spent $8 everyday and John spent $5 every day. Ali had $12 in the end while John had $24 in the end.
How much did each person have at first?

Step 1: Find the difference in the end

$24 – $12 = $12

Step 2: Find the difference for each day

$8 – $5 = $3

Step 3: Find the number of days

$12 ÷ $3 = 4

Step 4: Find the money each of them had at first

4 × $8 + $12 = $44

or 4 × $5 + $24  = $44 (Answer)

8. Grouping Concept

This is another common concept which needs you to group items together, followed by finding the total number of groups. 

Mark bought an equal number of shorts and shirts for $100. A shirt cost $8 and each pair of shorts cost $12. How much did he spend on the shirts?

Step 1: Group 1 shirt and 1 pair of shorts

8 + 12 = 20

Step 2: Find the number of groups

100 ÷ 20 = 5

Step 3: Find the amount spent on the shirts

5 × 8 = $40 (Ans)

9. Number x Value Concept

Under this concept, you multiply the number of units by the value of each unit to find the total value of 1 group. From here, you can find the total number of groups. 

The ratio of the number of 50 cents coins to 1 dollar coin is 3 : 1. The total value of the coins is $12.50. How many coins are there in total?

Step 1: Write down the ratio of 50 cents : $1

3 : 1

Step 2: Group three 50 cents coins and one $1 coin into 1 group

3 × 0.5 = $1.50

1 × 1 = $1

Step 3: Find the total value of 1 group

$1.50 + $1 = $2.50

Step 4: Find the number of groups

12.50 ÷ 2.5 = 5

Step 5: Find the total number of coins

5 × 4 = 20 coins (Ans)

10. Guess and Check / Assumption Concept

Assumption method is taught in many schools and it is a much faster and efficient way to do compared to guess and check. 

Miss Lee bought some pencils for her class of 8 students. Each girl received 5 pencils and each boy received 2 pencils. She bought a total of 22 pencils.  How many boys were there in the class?

Step 1: Start with an assumption (You can start with girls or boys).

Suppose there are 8 girls

Step 2: Find the total number of pencils

8 × 5 = 40

Step 3: Change your assumption

Suppose there are 7 girls, 1 boy.

Step 4: Find the total number of pencils

7 × 5 + 1 × 2 = 37

Step 5: Spot the pattern

40 – 37 = 3 (When the boys increase by 1, the total pencils decrease by 3)

Step 6: Find the total difference

40 – 22 = 18

Step 7: Find the number of boys

18 ÷ 3 = 6 boys (Ans)

Check your answer!
Number of girls = 8 – 6 = 2

Total pencils = 2 × 5 + 6 × 2 = 22 (Correct) 

11. Working Backwards Concept

The next type of common PSLE Math questions is “Working Backwards”. In this question, you are given the final value and you need to work backwards to find the starting value.

A bus left an interchange carrying some passengers with it.
At the first stop, 1/4 of the people in it alighted and 5 people boarded it.
At the 2nd stop, 1/2 of the people in it alighted and 20 people boarded the bus.
When it left the 2nd stop, there were 60 passengers in it.
How many passengers were there in the bus when it left the interchange?

Step 1: Find the number of people before the 2nd stop

60 – 20 = 40

40 × 2 = 80

Step 2: Find the number of people before the 1st stop

80 – 5 = 75

75 ÷ 3 × 4 = 100 people (Ans)

12. Simultaneous Equations Concept

In this type of PSLE Math questions, you need to form 2 equations to solve for 2 unknowns. 

Amy and Billy had a total of $400.  Amy spent 1/4 of her sum and Billy spent 2/5 of his. They then had a total of $255 left. How much did Amy spend?

Step 1: Let Amy’s money be 4 units, Billy’s money be 5 parts

A –> 4u

B –> 5p

Step 2: Form a first equation using their total amount of money at first

4u + 5p = 400   (Equation 1)

Step 3: Find the amount of money Amy and Billy have left

A –> 4u – u = 3u

B –> 5p – 2p = 3p

Step 4: Form a second equation using their total amount of money left

3u + 3p = 255

Step 5: Simplify the second equation to make the number of units the same as the first equation

3u + 3p = 255

u + p = 85 (Divide every term by 3)

4u + 4p = 340 (Multiply every term by 4)      (Equation 2)

Step 6: Use the first equation minus the second equation to find 1 part

4u + 5p = 400
–  (4u + 4p = 340)

–> 1p = 400  – 340 = 60

Step 7: Find 1 unit

85 – 60 = $25 (Ans)

13. Double If Concept

This is another type of PSLE Math questions which most students get stuck in. It involves 2 “ifs” which represent 2 scenarios. 

A farmer has some chickens and ducks. If he sells 2 chickens and 3 ducks every day, there will be 50 chickens left when all the ducks have been sold. If he sells 3 chickens and 2 ducks every day, there will be 25 chickens left when all the ducks have been sold.

a) how many ducks are there?

b) how many chickens are there?

Step 1: Write down the selling ratio for both cases

Case 1: Chicken : Duck = 2u : 3u

Case 2: Chicken : Duck = 3u : 2u

Step 2: Make the ratio of ducks to be the same as all the ducks are sold out in both cases.
Case 1 (Times 2 to both sides)

Chicken : Duck = 2u : 3u = 4u : 6u

Case 2 (Times 3 to both sides)

Chicken : Duck = 3u : 2u = 9u : 6u

Step 3: Form an equation using the chickens left

4u + 50 = 9u + 25

Step 4: Find 1 unit

5u = 50 – 25

1u = 5

Step 5: Find the total ducks

5 × 6 = 30 ducks

Step 6: Find the total chickens

5 × 4 + 50 = 70 chickens

Or

5 × 9 + 25 = 70 chickens

14. Equal Stage

There can be equal stage at first or equal stage in the end. You can use the “Model Drawing” method to solve this question.

Below is the coins question from the PSLE 2021 paper involving Equal Stage. 

Helen and Ivan had the same number of coins. Helen had some 50-cent coins and 64 20-cent coins.
Her coins had a total mass of 1.134 kg. Ivan had some 50-cent coins and 104 20-cent coins.
a) Who had more money and how much more?
b) Given that a 50-cent coin is 2.7 g heavier than a 20-cent coin, what is the mass of Ivan’s coins in kilograms?

Watch the video below on how to solve this question using Model Drawing.

15. Interval

Questions involving intervals test your child on finding the number of gaps between the items.

Take note that the number of gaps is 1 less than the number of items.

For example, there is 1 gap between 2 trees, 2 gaps between 3 trees, 3 gaps between 4 trees etc. 

Timothy participated in a marathon. There were drinking stations equally spaced along the route. Timothy took 48 min to run from the 1st drinking station to the 13th drinking station. Which drinking station would he reach after running for 120 min since the start of the marathon?

Number of gaps between the 1st and the 13th drinking stations = 12

Time taken to run the distance between 2 adjacent drinking station

= 48 ÷ 12

= 4 min

Number of gaps he would have run in 120 min

= 120 ÷ 4

= 30

30 + 1 = 31 (Ans)

15. Repeated Identity

This is another important concept which your child needs to know for PSLE Math.

Repeated Identity involves one of the items being repeated in the question. Your child needs to identify it and use the Ratio method and make it the same.

The number of adults to the number of children in a room is 5 : 6. There are twice as many boys as girls in the room. 

If there are 10 more adults than boys, how many people are there inside the room?

Adults : Children

= 5 : 6

Boys : Girls : Children

= 2 : 1 : 3

= 4 : 2 : 6 (Make the children the same)

5u – 4u = 1u

1u = 10

11u = 110 (Ans)

16. Replacement

This is a useful skill which your child can learn to apply.

It involves replacing one object with another object to solve the question. 

A school bus can either sit 24 adults or 36 children.

If it already has 12 adults on the bus, how many more children can it sit?

Adults : Children

= 24 : 36

= 2 : 3

= 12 : 18

By replacing the adults with children, there are 18 children on the bus.

36 – 18 = 18 (Ans)

Get a quick summary of all the types of PSLE Math Questions in our Free Mind-Maps Below. Click the download button to download.

“Finally… A Clear Summary of all the concepts which you need to know for PSLE Math” 

What number must be added in each term of fraction 7/31 so that it may become 5 17?

Which number must be added to each of the numbers 7 and 13 so that the resultant numbers will be in the ratio 2 3?

What should be added to 7/19/13 and 33 so that they are in proportion?

What must be added to each of the numerator and denominator of the fraction 7 by 11 to make it equal to 3 by 4?