### Technique #1: Repeated Identity

When given two or more ratios/fractions comprising three or more quantities, look out for the common quantity. This is the ** repeated identity**. Rewrite the ratios/fractions by changing the repeated identity to its Lowest Common Multiple (LCM).

__Example 1__

Alice has 3 : 4 of the number of sweets that Bob has. Bob has 2 : 5 of the number of sweets that Charlie has. Find the ratio of the number of sweets that Alice has to the number of sweets that Charlie has.

A |
B |
C |

3 | 4 | |

2 X 2 | 5 X 2 | |

3 | 4 | 10 |

The ratio of the number of sweets that Charlie has to the number of sweets that Bob has is 3 : 10.

### Technique #2: Ratio (Constant Total or Internal Transfer)

When a PSLE Math question involves one party giving to another, or two or more parties sharing a number of items, the total quantity remains unchanged. Rewrite the ratios, converting the total ratio units to its LCM.

__Example 2__

Martin has 5 times as many cards as Bob. After Martin has given Bob some of his cards, both of them have an equal number of cards. What fraction of his cards did Martin give to Bob?

Martin gave Bob 2/5 of his cards.

### Technique #3: Ratio (Constant Difference)

PSLE Math often tests students on their understanding that the difference of two quantities remains unchanged when (a) the question is about age; (b) giving away or gaining the same amount.

We can approach this type of problems by inferring that the difference between their ratio units remains constant. Rewrite the ratios, converting the difference in the ratio units or ages to its LCM.

__Example 3(a)__

4 years ago, the total age of Gina and her mother was 34 years. Gina will be 7 years old in 2 years’ time. In how many years will the ratio of Gina’s age to her mother’s age be 1 : 3?

The ratio of Gina’s age to her mother’s age will be 1 : 3 in 11 years.

__Example 3(b)__

Bernard had $820 and Dylan had $300. They each bought the same watch. Bernard then had 3 times as much money as Dylan. How much was the watch?

The watch was $40.

### Technique #4: Ratio (Constant Part)

When a PSLE Math question involves only one party gaining or giving away a number of items, the ratios between the parties at first and in the end will be different. To solve this type of questions, we need to infer that the other party, that did not gain or give away, **remains the same**. Rewrite the ratios, convert the before and after ratio units of the unchanged party to its LCM.

__Example 4__

The ratio of Charles’ marbles to Derrick’s marbles was 4 : 7. After Charles bought more marbles, the ratio of Charles’ marbles to Derrick’s marbles became 2 : 1. If their total number of marbles was 420 at the end, how many marbles did Charles buy?

Charles bought 200 marbles.

### Technique #5: Remainder Concept

This requires a strong understanding of fraction and percentage, in that any quantity on its own (whether it is the “at-first” amount or the remaining amount) is one whole or 100%.

We need to be able to identify “fraction … fraction **of the remainder**” as an indication of the Remainder Concept. Remember that percentages can be fractions in disguise, and that “of” in Math means “to multiply”.

__Example 5__

There are some Chinese, Malay and Indian pupils in a school hall, 2/9 of the pupils are Malay, 4/7 of the remaining pupils are Chinese and the rest are Indians. If there are 56 more Indian than Malay pupils, how many pupils are there altogether in the school hall?

There are 504 pupils altogether in the school hall.

### Technique #6: Sets

In some PSLE Math questions, we are given different quantities of two or more items, as well as the value of each item. When we encounter this, we can simplify it by grouping the items into sets, then finding the value of each set. The question usually gives us a total value, which we use to divide by the value of each set, to find the number of sets.

__Example 6__

Ethan earns $4 for every shirt that he sells. He gets a bonus of $16 for every 8 shirts that he sells. How many shirts must Ethan sell, if he wants to earn $924?

1 set (8 shirts) = 8 X $4 + $16 = $48

$924 ÷ $48 = 19 r $12

$12 ÷ $4 = 3 extra shirts

19 X 8 + 3 = 155 shirts

He has to sell 155 shirts to earn $924.

### Technique #7: Supposition

This one of the easiest techniques to master and bag that 3 or 4 marks! However, being the smart examination board that Singapore is, there have been slight modifications to this type of questions to trick the average students. We need to ensure we do not get tricked by the questions 🙂

__Example 7(a): Typical Non-Modified Question__

There are 26 twenty-cent coins and fifty-cent coins altogether. Given that the total amount of money is $8.20, how many twenty-cent coins are there?

There are 16 twenty-cent coins.

**Note**:

- Step 3 will be different if the question involves “penalty”. For example, $2 is awarded for every punctual delivery and $3 will be deducted for every late delivery. The difference between each item is now determined by
**adding**the two amounts instead of subtracting, because the difference in each punctual delivery and each late delivery is $2 + $3. The delivery guy loses the $2 award if the delivery is late, and has to pay a $3 penalty**on top**of it! - Step 4 will give you the answer for the quantity that was
**not**assumed in Step 1.

__Example 7(b): Modified Question to trick ‘average students’__

Arielle bought 34 pens and rulers altogether. Each ruler cost $0.40 and each pen cost $2.20. The **difference** between the total costs of the pens and the total costs of the rulers was $9.80. How many pens and rulers did she buy?

She bought 25 rulers.

### Technique #8: Reciprocals

This can be a fairly straightforward technique to apply when we are solving PSLE Speed and Rate questions.

__Example 8__

When turned on separately, Tap A can fill a bucket in 2 min while Tap B can fill the same bucket in 3 min. Find the time taken to fill the bucket if both taps are turned on.

Always determine the fraction of “work” that can be accomplished in one unit of time.

The time taken to fill the bucket if both taps are turned on is 72s.

### Technique #9: Units x Value (Multiple Model)

The Multiple Model is applied in situations whereby we are given (a) two or more known quantities and their respective values in ratio or fraction of each other, or (b) the ratio of two or more items and their known respective values.

__Example 9__

Janet spent $560 on some $5 vouchers and $10 vouchers. The ratio of the number of $5 vouchers to the number of $10 vouchers was 4 : 5. How many vouchers did she buy in all?

She bought 72 vouchers in all.

### Technique #10: Equal Concept (Equate Numerator)

This can appear in either PSLE Math Papers 1 or 2. We can identify this method when the question states that a fraction of one party is a fraction of another party.

__Example 10__

Half of Ethan’s allowance is equal to two-fifths of Gerald’s allowance. If they have $143 altogether, how much is Ethan’s allowance?

Ethan’s allowance is $78.

### Technique #11: Units & Parts (Simultaneous Equations)

This is a Lower Secondary technique to solve for the values of two unknowns using two equations. However, this type of questions have been appearing with increasing frequency in PSLE Math, usually with a 4 or 5 mark weightage in Paper 2. Simultaneous equations help us solve these PSLE questions easily and logically.

We use Units & Parts when the given ratios at first and in the end are different, and there is no way of comparing them using the ratio concepts of Constant Total, Part or Difference. This technique, when taught at the PSLE level, requires us to understand the following concepts:

- there is no way to solve for the numerical values of two or more unknowns in just a single equation;
- two unique equations are required to solve for two unknowns.

__Example 11__

Ms Cel and Ms Su baked some cookies in the ratio 5 : 6. After Ms Cel gave away some 40 cookies to her students, and Ms Su baked another 24 cookies, the ratio of Ms Cel’s cookies to Ms Su’s cookies became 1 : 3. How many cookies did Ms Su bake at first?

Ms Su baked 96 cookies at first.

### Technique #12: Bar Modelling

Some PSLE questions are best simplified with bar models, especially when we need to visualise the problem.

**Example 12**

Ethan read 164 pages more than half the number of pages in a novel on Friday. He read 12 pages more than 40% of the remaining pages on Saturday, then completed the novel when he read the last 48 pages on Sunday. How many pages were there in the book?

There are 528 pages in the book.

**Disclaimer:**

There are always other (more efficient or less efficient) methods to solve a (PSLE) Maths problem. AKLC does not endorse any method specifically. This article is merely a suggestion of some of the must-know techniques that well-prepared PSLE students have to equip themselves with.

**Ms Ancella Tan****Ms Nur Suriani**

**AKLC Online, Maths Mastery™**

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