#### 3 Simple Steps to Solve Commonly Tested Rate and Work Done Problem Sums

Sometimes, we argue and we wonder who the faster or more efficient worker is.

Bobby might say, “I took 2 and a half hours to build **nine tools in Roblox**!” while Billy might rebut with, “I took 1 hour and 15 minutes to build **four tools in Roblox**!”

To find out who the more efficient worker is, and to prevent them from getting into an argument, we simply use the **Rate Table** to make a comparison. In order to make a fair comparison, we need to **make the “time taken” the same** for both of them.

From the table, we can see that in the same amount of time, Bobby crafted 9 tools while Billy only crafted 8. Therefore, the faster and more efficient worker is Bobby.

The more challenging rate questions would involve finding the **combined rate of two or more objects** (e.g. how many tools can Billy and Bobby build together in an hour?) or questions that **involve a negative and positive input** (e.g. If Bobby builds 9 tools in an hour and Billy destroys 4 tools in an hour, how many hours would it take them to build 12 tools?).

# Let’s look at the following questions that are not as clear-cut as the example above and see for ourselves how the **Rate Table** helps us to solve the question quickly.

**Question 1**

https://www.pexels.com/photo/person-holding-paint-roller-on-wall-1669754/

**2021 SA1 P6 AI TONG SCHOOL**

Jimmy can paint a house in 10 days while Tom can paint the same house in 20 days. Jimmy started painting a house alone for 4 days and Tom joined him in painting the house on the 5th day. How many days did it take for the house to be completely painted?

**Step 1: Identify Question-Type**

Rate and Work Done

**Step 2: Select Rate Table Technique**

“Jimmy can paint a house in 10 days ...”

“Tom can paint the same house in 20 days ...”

**Step 3: Apply Common Units of Time in Table Technique**

The lowest common multiple of 10 and 20 is 20. Remember, we need to **make the “time taken” the same**

*in order to make a fair comparison.*

From the table, we see that in 20 days, Jimmy can paint 2 houses by himself by Tom can only paint 1 house by himself.

*“Jimmy started painting a house alone for 4 days...”*

Since they did not start painting together on the same day, we need to find out how much of the house was painted by Jimmy during the* first four *days.

Let’s find the fraction of a house that is painted by Jimmy alone for 4 days.

When Tom joined Jimmy in painting the house, **⅖ **of the house had already been painted. This means that they had **⅗ **of the house left to paint. Let’s refer to the same table below to find out their rate of work when both of them are working together.

From the table, we see that in 20 days, Jimmy can paint 2 houses by himself while Tom can paint 1 house. Together, they can paint a total of 3 houses in 20 days.

Jimmy painted alone for *4 days at first,* before Tom joined him. Together, they took *another *4 days to finish painting the rest of the house. In total, it took *8 days* for the house to be completely painted.

**Question 2**

https://www.pexels.com/photo/close-up-photo-of-water-drop-2583028/

**2021 SA1 P6 CHIJ OUR LADY OF GOOD COUNSEL**

Tap A can fill a container with a capacity of 310 ml of water in 2 minutes. Tap B can fill a container with a capacity of 585 ml of water in 3 minutes. At this rate, how long will it take to fill a container of 3.5 litres when both taps are turned on at the same time?

**Step 1: Identify Question-Type**

Rate and Work Done

**Step 2: Select Rate Table Technique**

“Tap A can fill a container with a capacity of 310 ml of water in 2 minutes...”

“Tap B can fill a container with a capacity of 585 ml of water in 3 minutes...”

**Step 3: Apply Common Units of Time in Table Technique**

The lowest common multiple of 2 and 3 is 6. Remember, we need to **make the “time taken” the same**

*in order to make a fair comparison. Hence, after restating, this is what we have:*

From the table, we see that in 6 minutes, 930 ml of water flows out of Tap A while 1170 ml of water flows out of Tap B. Together, 2100 ml of water flows out from both taps in 6 minutes.

*“At this rate, how long will it take to fill a container of 3.5 litres when both taps are turned on at the same time?”*

**3.5 litres = 3500 ml**

When both Tap A and Tap B are turned on at the same time, it would take them 10 minutes to fill a container with a capacity of 3.5 litres.

After mastering this Rate Table technique, most students find it fun to fill in the table and find out for themselves which tap has water flowing out of it at a faster rate, or how much time is needed to paint a house, or who can type more words in a given amount of time! The scenarios are endless!

For this technique, students need to remember to **make the “time taken” the*** same*

**in order to make a fair comparison, and check if the objects in the question are making a positive or negative input.**