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The phrase "approximately the same weight as" is correct and usable in written English.
It can be used when comparing the weights of two objects or entities that are similar in mass but not identical.
Example: "The package weighs approximately the same weight as a standard laptop."
Alternatives: "roughly equivalent in weight to" or "about the same mass as".
Exact(3)
"Every industry is represented at approximately the same weight as [in] the Russell 1000," said Just Capital's Hernando Cortina, referring to the best-known index fund tracking the largest publicly traded companies.
That is, according to a computer scientist at UC Berkeley, who calculated in 2011 that a 4GB ebook reader filled with 3,500 ebooks weighs a billionth of a billionth of a gram more than if it were empty of data - a difference that is approximately the same weight as a molecule of DNA.
Since one proton has approximately the same weight as one neutron, and the atomic number represents the number of protons, we can simply subtract the number of protons from the total mass.
Similar(57)
Because the late Permian cenospheres are approximately the same size and likely about the same weight as the smallest particles of volcanic ash, they could have easily risen to an altitude of about 20 kilometers in the atmosphere and then been swept around the world by jet stream winds.
Luxembourg would have the same weight as China.
Each lead filled bronze the same weight as an astronaut.
The average cloud is the same weight as 100 elephants.
I'm approximately the same weight.
"Google needs to give its own products the same weighting as those of others".
If the system delay is low, packets from different priority classes have approximately the same weights.
A hack fight is a pickup match made on the spot between any two men who have birds approximately of the same weight that want to bet money.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com